They need to be dealt with in fully developed scenes. There is nothing quite as dissatisfying as having a major problem in the story be dealt with off-screen. The resolution should feel inevitable, even if it surprises. The resolution of your story can be thought of the ultimate surprise, the revelation of the central mystery. Even more than the smaller mysteries and surprises, the primary resolution of your story should be the logical coming together of facts and events known to the reader.
Inevitability comes, like suspense, from foreknowledge.
Don't dangle threads without dealing with them. Lastly, when you raise a question or a problem in a story, do so with the intention of eventually dealing with it before the end of a story. Dealing with it can be as simple as an acknowledgment that the problem isn't going to be solved within the space of the story, but the acknowledgment needs to be there or the reader will feel as if the writer simply forgot about it. By doing this, you have evolved from a character dealing with a problem, to a character dealing with a worse problem that's directly and causally linked to the first.