Quests are special missions which can be found under the ‘Quests’ button in the Base. Quests reward you with guaranteed loot of a specified type when fully completed.
The type of the loot is always specified in the name of a Quest. A “Find Recruits” Quest, for example, gives Recruits as its final reward. This makes Quests great for finding resources you might otherwise have trouble finding.
Different Quests appear randomly, though some are more common than others. Quests are divided into three rarities: Common, Uncommon and Rare Quests. The rarer the Quest, the higher the prizes are for beating it, but so is its difficulty.
Once you have beaten all available Quests, you might not be immediately offered new ones to complete. There is no need to worry, however - new ones will become available after a while. Quests always change daily, so if you encounter Quests you do not wish to complete, you only need to wait for the next ones.
Rare Quests are rarely appearing Quests that feature highly difficult battles but also equally great rewards. Completing Rare Quests is a great way of finding rare Ascension Materials you might have trouble finding.