In higher organisms, all cells share the same genome, but every cell expresses only a limited and specific set of genes that defines the cell type. During cell division, not only the genome, but also the cell type is inherited by the daughter cells. This intriguing phenomenon is achieved by a variety of processes that have been collectively termed epigenetics: the stable and inheritable changes in gene expression patterns. This article reviews the extremely rich and exquisitely multi-scale physical mechanisms that govern the biological processes behind the initiation, spreading and inheritance of epigenetic states. These include not only the changes in the molecular properties associated with the chemical modifications of DNA and histone proteins, such as methylation and acetylation, but also less conventional changes, typically in the the physics that governs the three-dimensional organization of the genome in cell nuclei. Strikingly, to achieve stability and heritability of epigenetic states, cells take advantage of many different physical principles, such as the universal behavior of polymers and copolymers, the general features of dynamical systems, and the electrostatic and mechanical properties related to chemical modifications of DNA and histones. By putting the complex biological literature under this new light, the emerging picture is that a limited set of general physical rules play a key role in initiating, shaping and transmitting this crucial "epigenetic landscape". This new perspective not only allows to rationalize the normal cellular functions, but also helps to understand the emergence of pathological states, in which the epigenetic landscape becomes dysfunctional.