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Losing is fun!
Either way, it keeps you busy.
There is no internal end point, single goal, final Easter egg or "You Win!" announcement in Dwarf Fortress. Therefore, eventually, almost every fortress will fall. The only ones that don't tend to be very conservative and very boring—and what fun is that? Therefore, DF = losing ∧ DF = fun ⇒ losing = fun, and that's okay! It's a game philosophy, so embrace it, own it, and have fun with it!
Most new players will lose their first few forts sooner rather than later; when you lose a fortress, don't feel like you don't understand the game. Dwarf Fortress has a steep learning curve, and part of the process (and fun!) is discovering things for yourself. However, this Wiki serves as an excellent place to speed up the learning process.
 Autopsy, or why your fortress died
Various common things can cause the death of a fortress. Let's examine some together...
 Dwarf vs. Wild
Sometimes the wilds take you out.
 Local Wildlife
Goblins aren't the only creatures that want you dead. The obvious threats aside, some creatures with benign natures, names, or descriptions can be surprisingly deadly. A sudden wildlife attack can quickly cripple or destroy an unprepared fortress. Before you unpause a new game for the first time, hit nits, and switch to the "Others" tab to see what's sharing your map. Learn to do this regularly—new creatures will frequently migrate onto your map and then off again to be replaced by others.
Consider arming up and thinning out any predictable threats.
 Underground Life
Underground life can be even more dangerous than surface life. Dig down to a cavern, and expect to be fending off hordes of smaller, weaker creatures as well as larger, more solitary creatures like giant cave spiders and blind cave ogres. Arming up helps a lot, as there is usually only a small entrance they can get in by. A row of cage traps is exceptionally powerful there. Underground Forgotten Beasts are a later stage hazard—and one that cage traps will offer very little protection against. Even if all the other creatures in the cavern are stopped by your cage traps, don't allow yourself to become complacent.
 Magma and its denizens
Magma by itself is dangerous enough to destroy a fortress that fails to contain it properly, but magma sources are also home to many dangerous creatures that can destroy buildings, kill even the most skilled soldiers, and spread destruction throughout your fortress. While careful designs using fortifications, magma-safe floor grates, and magma-safe bridges can mitigate much of the danger, magma remains a prime source of fun even in experienced fortresses.
 Evil Biomes
Evil surroundings are more inhospitable than ever. If it's not the zombies that kill all your dwarves, then it's their freshly animated body parts hungry for revenge. Or vile weather that makes dwarves caught in it vomit to death. Or, worst of all, an evil cloud that drifts by your settlement and starts a zombie plague in your population.
When embarking on an evil biome, it's a fairly good idea to skip a meat industry of any kind. Though meat obtained from butchering doesn't rise up, hair, skin and bones are fully capable of turning to kill you - and bringing hunters in your party is near-suicidal. Embarking with a military is a must, if you want to stand any chance against the zombies. Of course, if things are really bad outside (for example, the aforementioned thralling clouds), you're best off just sealing yourself inside forever and never turning back. As with cavern wildlife, a row of cage traps is a great boon to the safety of your fortress.
 Dwarf vs. You
Sometimes you bring it on yourself.
 No Food
A serious danger, generally in the more inhospitable climates, is the loss of your dwarves due to starvation. As dwarves begin to starve, they will become hungry, then starving. This will cause them first to slow down all work, and then to become very unhappy. When they die, their friends will become upset and will become even more unhappy, potentially causing the remainder of your fortress to break out in a terminal hissy fit.
Don't forget your alternative sources of food. If your farms aren't doing the job and a trade caravan is months away, try butchering your domestic animals, gathering plants, hunting local wildlife, and cooking any excess alcohol.
 No Booze
Equally as bad is no alcohol, which dwarves require to be happy and productive. Some alcohol can be acquired from caravans, but not enough for an entire fort until the next caravan arrives. You must gather or farm certain plants to then brew those in a still with an empty barrel—it's just part of being a dwarf. Be sure to make lots of barrels. Often a shortage of barrels is just as bad as a shortage of beer. A dwarf would rather die than lower themselves to drinking from a mug (though it doesn't stop them being produced by the tonne in your workshops).
Sometimes you just can't help but take a drink, and when you think it's safe, WHAM. Watch out for giant sponges...
 No water
Healthy dwarves will not die of thirst as long as they have alcohol, which in the current version can be brewed without the use of water. However, injured dwarves must be given water, not alcohol, or they will die of dehydration.
Rain will refill stagnant pools of water slowly. In a hot climate, this may evaporate almost immediately. What's more, if the map is in a dry climate, such as a desert (hot or cold), then there can be long periods of time with no water anywhere - in extreme cases, none ever. Snow will not refill pools, so you can also have a lack of water in very cold climates. Also, if weather has been turned off in the init.txt file then there will be no rain and no water will accumulate, though it may be there at the beginning of the game.
 Too Much Water (aka "Flooding accidents")
The opposite side of the dehydration spectrum is having too much water. Remember that water can flow in 10 directions (the 8 horizontal ones as well as down, and up to the level of its source.)
If your fortress is beginning to flood from sourced water, abandon all of the levels the water can reach immediately—create a civilian alert and order your dwarves to a burrow upstairs. You will never be able to recover those areas unless you can manage to pump out the water faster than it floods in, which can take over a year or two of game time to establish a functioning automated pump system. Generally, a flooding accident spells doom for your fortress.
If the accident is causing your mine shafts to flood (those fishy diagonal flows into downstairs on the level below), you can sometimes save the dwarves that are working inside it: designate the highest level they can reach before the water reaches it with your civilian burrow. Try to dig your way up from there, since the water will take some time to fill the lower floors, and keep updating your burrow definition to the higher floors. Saving your valuable Legendary +5 Miners (and their picks) could be vital to your fortress.
Sometimes, a fortress is flooded with magma. This is even more fun, and even harder to recover from. Magma doesn't rise as aggressively (via pressure) as water, and can be pumped out with magma-safe equipment. Read up on it. Good luck.
 Inability to mine ("Diggor Mortis")
i.e., no picks.
Diggor Mortis: when a Dwarf with a pickaxe decides that digging where they shouldn't is a bloody good idea.
Simply put, you need picks to mine ore, which is then smelted to make metal for items like more picks. If you are careless (or ignorant) of how to dig safely, and your miners create a collapse or flood and their equipment gets lost/destroyed/unrecoverable, and you have no materials to make more picks, you will be at a severe handicap until the problem is solved. Any dwarf can be given the mining labor, but without a pick they can do nothing. There is no way to get new metals or stone (clay excluded) for any purpose (except from foreign traders and smelting other items) nor any way to dig new rooms/tunnels unless you have picks.
If you have ore or bars to create a weapons-grade metal (other than silver), and a forge (and smelter if you need one), you can create new picks and continue. In a pinch, you can even melt other metal objects for metal. You might get lucky with a caravan - elves never carry picks, but humans sometimes have bronze ones, and dwarves generally bring some along. If the first dwarven caravan doesn't bring any, you can try to keep your fortress running long enough to request additional picks from your outpost liaison, who will arrive with the next dwarven trade caravan in a year. Or you can abandon and try again.
Averting this fate is simple: stockpile at least one additional pick at the first possible opportunity, or some of the raw material to make more, and away from current digging operations.
See also: Arms industry
 Dwarf vs Society
Sometimes your traditions and morals conspire. Hell Is Other Dwarves?
 General Unhappiness
If there is little in a fortress to give your dwarves happy thoughts and enough to give them unhappy thoughts, then your dwarves will start to throw tantrums, grow melancholy, and/or cause general chaos. In extreme (but sadly not rare) examples, this can lead to a tantrum spiral and the loss of the entire fortress. Immigrants are typically married, often with children in tow, and are primed and ready to start pumping out even more babies. This means that the loss of one dwarf is likely to lead to a whole family of very unhappy dwarves, and potentially a tantrum spiral. Unhappiness is more likely to occur if your fortress is suffering other kinds of downfall as well, so try to keep all the bases covered.
Another problem is if you don't have a justice system in place at the time of a spiral and manage to recover. If you later implement the justice system, the hammerer may kill the former tantrum throwers, starting another tantrum because of their deaths.
While it doesn't happen often, those Nobles' mandates can occasionally cause your fortress to come to a halt, especially if they decide to imprison a dwarf that's fairly essential to your fortress, or if they for some reason decide that a dwarf deserves a hammering. This can lead to your dwarf's death, which, as noted above, can cause a tantrum spiral.
Then, there's also points at which, when trying to fulfill a mandate, you run into some rather nasty problems. This could lead to any of the other problems, such as a cave in killing your miner (If you're not paying attention!) running into HFS, or accidentally mining out a new tunnel to the surface that you don't close, and lets in a horde of goblins.
 Supernatural Rogues
With DF2012 have come the additions of several night creatures that can potentially undo a fortress from within. A werebeast raid can not only bring trouble if you're lacking of a military, but its cursed bite can make more of its kind from your population. If you see a civilian survive a werebeast attack, you might be in for a surprise when he suddenly turns into a ravenous monster at the full moon and starts killing everything around him.
Vampires are a much more insidiously dangerous threat, as they hide themselves among the peasantry and may kill dozens of dwarves without being caught—and if they do, they might succeed at framing an essential member of your fortress into getting the hammer. As can be seen, this can easily lead into a tantrum spiral.
 Dwarf vs Goblins, Humans, Elves...
Seizing goods from a friendly caravan will often lead to large shipment of fun next time the traders visit. Remember, it's not paranoia if they are out to get you.
Goblin and elven ambushers alike will infiltrate your map and charge into your fortress after they are discovered. They will retreat after suffering enough casualties or waiting around several seasons. Elves still arrive with caravans, and goblins can attack at any time. Even if your dwarves do not venture onto the surface, caravans will eventually trigger the ambushes. At least, caravans and sieges/ambushes have the amazing tendency to occur at the end of each season.
Sieges can be quite devastating to a fortress, but unlike most of the other ways of losing, they are unlikely to occur early on, even if you do something stupid to piss off another civilization.
Should hosts of goblins besiege your gates and drive your peasantry inside, trolls beat down your doors and force you to seal off from the outside world, you may have already lost the game. Even if you have built an utterly impenetrable fortress with drawbridges and moats, a sieging army may stick around for a long time. Although a dwarven fortress can be made self-contained, with crops, metal and fuel readily available, underground wood source and your own livestock, a fortress may not be able to sustain such a state indefinitely.
For example, trade with the outside world has now been shut off, leaving you only what ores are on your map for the production of mandated goods. In the (very) long run even those will run out. This can result in a breakdown of social order if you do not prevent your Hammerer from killing or maiming your dwarves. Shell, bone and leather commonly acquired by hunting and fishing need to be supplied by previously established livestock and access to suitable water. If these resources are no longer available to your workers, moody craftsdwarves will be driven into suicide or worse. Rotten vermin corpses begin to heap in your food supply, forcing you to dump these into inside refuse piles, generating miasma. Better build indoor refuse piles away from trafficked areas. Unless an interior water supply was established (or you find water in caverns) your wounded will die of dehydration.
With all these critical industries unproductive, dwarves dying, and friends mourning over the rotting heaps of slain loved ones, it's important to remember your dwarves have nothing to do but throw funeral receptions, hold grief counseling sessions and host the occasional keg stand. This means they've all become one big happy family of friends, manically depressed from the loss of any dwarf.
In short, the attacking army can simply wait until your dwarves emo themselves to death.
 Hidden Fun Stuff
Hidden fun stuff awaits those who delve too deeply and too greedily. Due to the concentrated amounts of fun, HFS has been the downfall of many a fortress.
 Dwarf vs. Player
Mod too much and the human caravan that comes in could spontaneously combust. Or just crash your game.
 Dwarf vs. Computer
Fall victim to the Catsplosion and even the mightiest computer may lose FPS to the point where the game becomes virtually un-playable. Or get too many Dwarves and pets in your fortress and the computer may crash. Problem can be solved, at least for the short term, by changing the priority of Dwarf Fortress in init.txt.
 Dwarf vs. Boredom
Sometimes a fort's downfall comes not from fire and Fun, but by the player deciding that enough's enough, and it's time to give in.
 Have I Won Yet?
Even the most well-managed of forts will eventually fall. And if they don't, then what? Once you've fended off goblins, bronze colossi and other fun things, what's left to challenge your might defenses and legendary champions?
Eventually boredom takes its toll and you either hit abandon, or ease up on the impregnable defenses and effectively engineer your own doom.
 The Tantrum Fizzle
So your dwarves are too busy punching each other to do any work, and the casualties are mounting while the goblins just point and laugh. Think the final chapter of your fort's saga will be a short one? Don't be so sure. Sometimes a little old-fashioned anarchy is enough to cripple a fortress, but not destroy it.
With good defenses and enough supplies, a well-managed fortress can run quite a while without any player input. And when you find this out the hard way, a few in-game months of fistfights might not be entertaining enough to offset the fact that you're not able to actually DO anything. If things don't calm down (or explode hilariously) soon enough, that "abandon fort" option will look awfully tempting...