Hutchinson-Gilford progeria syndrome (HGPS) is an extremely rare premature aging disease presenting many features resembling the normal aging process. HGPS patients die before the age of 20 years due to cardiovascular problems and heart failure. HGPS is linked to mutations in the LMNA gene encoding the intermediate filament protein lamin A. Lamin A is a major component of the nuclear lamina, a scaffold structure at the nuclear envelope that defines mechanochemical properties of the nucleus and is involved in chromatin organization and epigenetic regulation. Lamin A is also present in the nuclear interior where it fulfills lamina-independent functions in cell signaling and gene regulation. The most common LMNA mutation linked to HGPS leads to mis-splicing of the LMNA mRNA and produces a mutant lamin A protein called progerin that tightly associates with the inner nuclear membrane and affects the dynamic properties of lamins. Progerin expression impairs many important cellular processes providing insight into potential disease mechanisms. These include changes in mechanosignaling, altered chromatin organization and impaired genome stability, and changes in signaling pathways, leading to impaired regulation of adult stem cells, defective extracellular matrix production and premature cell senescence. In this review, we discuss these pathways and their potential contribution to the disease pathologies as well as therapeutic approaches used in preclinical and clinical tests.