Quests are not being completed.

It's hard to give advice when you don't provide detailed, step-by-step information about what you've done. If you're not using scripts, you need to set the quest type in the MapPropertiesTree and add parameters to the tree (Parameters -> Add). The parameters themselves contain the details, such as the ID of the artifact that needs to be obtained (and possibly the name of the hero who should receive it?). More details can be found in the map editor manuals that come with the game. Also, pay attention to the flags, such as IsHidden and others. Usually, their names are self-explanatory. A single incorrect value (false instead of true, or vice versa) can seriously break the quest logic.