I am only a beginner in the field, but I would like to have at least a few examples of open problems in deformation theory to give me an idea of the actual status of the theory, and to help guide my learning as well.

I am interested both questions about specific deformation problems and the general theory (by which I mean problems starting with: let $F$ be a deformation functor...)

It would be nice to have an answer for every open problem posted, possibly with some details and references.