Genetic material in the nucleus governs mechanisms related to cell proliferation, differentiation, and function. Thus, senescence and aging are directly tied to the change of nuclear function and structure. The most important mechanisms that affect cell senescence are: (i) telomere shortening; (ii) environmental stress-mediated accumulation of DNA mutations, and (iii) the intrinsically encoded biological clock that dictates lifespan events of any particular cell type. Overall, these changes lead to modification of the expression of genes that are responsible for: (i) organization of the nuclear structure; (ii) integrity of transcriptionally inactive heterochromatin, and (iii) epigenetic modification of chromosomes due to DNA methylation and/or histone modifications. These aging-related nuclear alterations do not only affect somatic cells. More importantly, they affect stem cells, which are responsible for proper tissue rejuvenation. In this review, we focus on epigenetic changes in the chromatin structure and their impact on the biology and function of adult cells as they age. We will also address aging-related changes in a compartment of the most primitive pluripotent stem cells that were recently identified by our team and named ‘very small embryonic/epiblast-like stem cells’.