Quests
NPC given quests were introduced in version 3.40 and range from collecting branches to fighting robbers. Quests are received by either chatting (c) to the quest giving NPC, or, if the quest giver believes his quest urgent, they might approach the adventurer themselves. Information about all received quests, both finished and unfinished, can be found by pressing F2. Quests and quest related areas will also marked in the adventurers world map accessed by pressing F6. There's also always a reward for completing a quest.
General tips
All quest related discussion options are under the option "Greet".
Many quests have an area for the adventurer to search for whatever it is that was requested. In these situations adventurers are reminded that climbing a tree (Alt+l) gives a good view of the surrounding area, and that one should look further than one's immediate surroundings (Ctrl+- or scrolling with mousewheel).
If one expects to find tracks in the quest related area, they can also be looked for (Alt+t) from the zoomed-out map. Finding the tracks this way is not guaranteed however.
If one is searching for a wounded adventurer's shelter, a good strategy would be to track (Alt+t) the animal in the quest description from the zoomed out map. Track each tile on the wilderness map which fits the adventurer's description. Once one has found tracks of the animal (bear, wolf, etc.), they should enter that local tile and climb a nearby tree (Alt+l) and look around locally (Ctrl+-). Only once the shelter is discovered locally, it will appear on the wilderness map.
Quests and quest specific information
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I tried just to walk to the village on the zoomed-out map, but I lost him. Now I try walking him there on the zoomed-in map (which may take a while), but that seems to work.
A wounded adventurer has crawled to a village and is resting and recuperating there. If villagers are asked "How is it going" they will tell that there's an injured adventurer in the village. This quest is received by talking to the wounded man who will tell you that he was attacked by a bear or a wolf while collecting firewood near his shelter and that his belongings, including a family heirloom which he'd like back, were left by the shelter.
The general tips work very well if one has trouble finding the shelter. The beast may still lurk around and following its tracks backwards may help locate correct area. The belongings are usually scattered within on tile of the shelter making spotting it easier. As a reward, you may get a lesson (4%) in one combat skill, or sometimes the location of a treasure.
A villager will tell you that someone's been stealing prey from his bird traps and wants you to ask an other villager about a spell to catch the thief and then cast it. You'll need to give the other villager a whole bird carcass (freshness doesn't matter) so catch a bird but don't skin or butcher it. You'll also need three feathers of a stolen catch and for that you'll need to do the most annoying part of this quest: locate the trap which is only marked by a couple of feathers lying around with an unset loop snare. The place is very easy to miss.
Once the feathers have been acquired, the spell can be cast in a village sauna or a base rock after either of these has a burnt out fire (you can check by looking with l, this will also tell if a gray ground square is base rock). The bird thief will then come to confess to the quest giver, so talk to him after a day or two to receive your reward: a lucky paw.
When asking a villager how's it going they'll tell you that a villager has been very busy lately. When asking the busy busy busy villager for a quest, they'll ask you to gather stones and take them to the sauna. If one is uncertain which building is the sauna, it'll have the sauna scoop in it. Reward: a few squirrel hide's worth of goods. These will be used when next time trading with the village.
When asking a villager how's it going they'll tell you that a villager has been very busy lately. When asking the busy busy busy villager for a quest, they'll ask you to gather branches and drop them anywhere in the village. Reward: a few squirrel hide's worth of goods. These will be used when next time trading with the village.
When asking a villager how's it going they'll tell you that the sage/shaman could use some help. When talking to the sage, he'll ask the player to collect some herbs for him. These herbs may be unknown to the player and even poisonous, but the wise old man who's been eating weird plants all his life must know what he's doing, right? If you are having trouble locating or identifying the correct herb, the plants page may be of use. For a reward, you'll receive some all-healing ailment that helps with injuries when boiled. It's uncertain whether this ailment helps with poisoning or deadly poisoning so experimentation is player's own responsibility.
I spoke to one NPC who had lost a sheep and he said the Shaman knew more. I spoke to the Shaman who told me to search for the sheeps tracks and light a fire of spruce twigs to make the sheep appear. I randomly wandered about the quest area in wilderness view using tracking until I found some tracks (it took 12 days). I lit a fire on the last track I could find and then caught a glimpse of a sheep through the trees. I didn't have to tether him with rope (he bleated in disagreement) he just followed me all the way to the village, even when I slept he stayed close by. I didn't dare go back to wilderness view to travel so stayed on local map in case I lost the sheep.
Upon entering a village you have previously visited, you may be automatically engaged in dialogue (before loading the local area). The villager will warn you of robbers in the region and direct you to speak to another villager. After speaking to the second villager, they will give you a rough location of the robbers with a warning that they might have moved on. They also directs you to speak to the original villager about it. The first villager who spoke to you is where you may abandon or turn in the quest. The robbers will not be present on the world map.
Robbers seem to spawn randomly when entering a local map within the specified region. I apparently spawned them without ever seeing them. So look around a bit in the local tiles!