The nucleus of a differentiated cell can be reprogrammed to a totipotent state by exposure to the cytoplasm of an enucleated oocyte, and the reconstructed nuclear transfer embryo can give rise to an entire organism. Somatic cell nuclear transfer (SCNT) has important implications in animal biotechnology and provides a unique model for studying epigenetic barriers to successful nuclear reprogramming and for testing novel concepts to overcome them. While initial strategies aimed at modulating the global DNA methylation level and states of various histone protein modifications, recent studies use evidence-based approaches to influence specific epigenetic mechanisms in a targeted manner. In this review, we describe—based on the growing number of reports published during recent decades—in detail where, when, and how manipulations of the epigenome of donor cells and reconstructed SCNT embryos can be performed to optimize the process of molecular reprogramming and the outcome of nuclear transfer cloning.