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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.

20 Open questions about deformations of compactifiable manifolds