In eukaryotic cells, the nucleus houses the genomic material of the cell. The physical properties of the nucleus and its ability to sense external mechanical cues are tightly linked to the regulation of cellular events, such as gene expression. Nuclear mechanics and morphology are altered in many diseases such as cancer and premature ageing syndromes. Therefore, it is important to understand how different components contribute to nuclear processes, organisation and mechanics, and how they are misregulated in disease. Although, over the years, studies have focused on the nuclear lamina—a mesh of intermediate filament proteins residing between the chromatin and the nuclear membrane—there is growing evidence that chromatin structure and factors that regulate chromatin organisation are essential contributors to the physical properties of the nucleus. Here, we review the main structural components that contribute to the mechanical properties of the nucleus, with particular emphasis on chromatin structure. We also provide an example of how nuclear stiffness can both impact and be affected by cellular processes such as DNA damage and repair.