The integrity of the nuclear lamina has emerged as an important factor in the maintenance of genome stability. In particular, mutations in the LMNA gene, encoding A-type lamins (lamin A/C), alter nuclear morphology and function, and cause genomic instability. LMNA gene mutations are associated with a variety of degenerative diseases and devastating premature aging syndromes such as Hutchinson-Gilford Progeria Syndrome (HGPS) and Restrictive Dermopathy (RD). HGPS is a severe laminopathy, with patients dying in their teens from myocardial infarction or stroke. HGPS patient-derived cells exhibit nuclear shape abnormalities, changes in epigenetic regulation and gene expression, telomere shortening, genome instability, and premature senescence. This review highlights recent advances in identifying molecular mechanisms that contribute to the pathophysiology of HGPS, with a special emphasis on DNA repair defects and genome instability.