It is an open problem whether there are only finitely many diffeomorphism types of projective Calabi-Yau 3-folds.
Kawamata--Namikawa developed log deformation theory of normal crossing Calabi-Yau varieties.
As an application of their result, one can construct examples of Calabi-Yau manifolds by smoothing SNC varieties.
In this talk, I will explain how to construct examples of non-Kähler Calabi-Yau 3-folds which are topologically unbounded.
If time permits, I will also explain an example of involutions on a family of K3 surfaces which do not lift biregularly to the total space.
This is based on joint work with Kenji Hashimoto.