To date biomedicine and pharmacology have required generating new and more consummate models. One of the most perspective trends in this field is using induced pluripotent stem cells (iPSCs). iPSC application requires careful high-throughput analysis at the molecular, epigenetic, and functional levels. The methods used have revealed that the expression pattern of genes and microRNA, DNA methylation, as well as the set and pattern of covalent histone modifications in iPSCs, are very similar to those in embryonic stem cells. Nevertheless, iPSCs have been shown to possess some specific features that can be acquired during the reprogramming process or are remnants of epigenomes and transcriptomes of the donor tissue. These residual signatures of epigenomes and transcriptomes of the somatic tissue of origin were termed “epigenetic memory.” In this review, we discuss the “epigenetic memory” phenomenon in the context of the reprogramming process, its influence on iPSC properties, and the possibilities of its application in cell technologies.