The Hutchinson–Gilford progeria syndrome (HGPS) is a premature aging disease caused by mutations of the LMNA gene leading to increased production of a partially processed form of the nuclear fibrillar protein lamin A – progerin. Progerin acts as a dominant factor that leads to multiple morphological anomalies of cell nuclei and disturbances in heterochromatin organization, mitosis, DNA replication and repair, and gene transcription. Progerin-positive cells are present in primary fibroblast cultures obtained from the skin of normal donors at advanced ages. These cells display HGPS-like defects in nuclear morphology, decreased H3K9me3 and HP1, and increased histone H2AX phosphorylation marks of the DNA damage loci. Inhibition of progerin production in cells of aged non-HGPS donors in vivo increases the proliferative activity, H3K9me3, and HP1, and decreases the senescence markers p21, IGFBP3, and GADD45B to the levels of young donor cells. Thus, progerin-dependent mechanisms act in natural aging. Excessive activity of the same mechanisms may well be the cause of premature aging in HGPS. Telomere attrition is widely regarded to be one of the primary hallmarks of aging. Progerin expression in normal human fibroblasts accelerates the loss of telomeres. Changes in lamina organization may directly affect telomere attrition resulting in accelerated replicative senescence and progeroid phenotypes. The chronological aging in normal individuals and the premature aging in HGPS patients are mediated by similar changes in the activity of signaling pathways, including downregulation of DNA repair and chromatin organization, and upregulation of ERK, mTOR, GH-IGF1, MAPK, TGFβ, and mitochondrial dysfunction. Multiple epigenetic changes are common to premature aging in HGPS and natural aging. Recent studies showed that epigenetic systems could play an active role as drivers of both forms of aging. It may be suggested that these systems translate the effects of various internal and external factors into universal molecular hallmarks, largely common between natural and accelerated forms of aging. Drugs acting at both natural aging and HGPS are likely to exist. For example, vitamin D3 reduces the progerin production and alleviates most HGPS features, and also slows down epigenetic aging in overweight and obese non-HGPS individuals with suboptimal vitamin D status.