Extra extension properties of equidimensional holomorphic mappings: results and open questions

Holomorphic (nondegenerate) mappings between complex manifolds of the same dimension are of special interest. For example, they appear as coverings of complex manifolds. At the same time they have very strong "extra" extension properties in compare with mappings in different dimensions. The aim of this paper is to put together the known results on this subject, give some perspective on the general strategy for future progress, prove some new results and formulate open questions.