  Deformation theory of complex manifolds is a classical subject with recent
new advances in the noncompact case using both algebraic and analytic methods.
  In this note, we recall some concepts of the existing theory and introduce
new notions of deformations for manifolds with boundary, for compactifiable
manifolds, and for $q$-concave spaces. We highlight some of the possible
applications and give a list of open questions which we intend as a guide for
further research in this rich and beautiful subject.
