Inside cell nuclei in eukaryotic organisms, genomic DNA is packaged into arrays of nucleosomes. Each fully wrapped nucleosome consists of 147 base pairs of DNA wrapped around a histone octamer core. The resulting complex of DNA with histones and other proteins forms a multi-scale structure called chromatin. At the most fundamental level of chromatin organization, arrays of nucleosomes form 10-nm fibers that are thought to resemble beads on a string. Chromatin fibers fold into higher-order structures which ultimately make up functional chromosomes. Depending on the organism and the cell type, 75-90% of genomic DNA is packaged into nucleosomes. The question of how various cellular functions such as gene transcription are carried out on the chromatin template is a biological puzzle that can be treated using tools from statistical mechanics. In this talk, I will discuss recent advances in understanding fundamental biophysical mechanisms of chromatin equilibrium and non-equilibrium dynamics. In particular, I will demonstrate that in baker's yeast, neighboring nucleosomes invade each other's territories through DNA unwrapping and translocation, or through initial assembly in partially wrapped states. Thus, the classic "beads-on-a-string" picture of well-positioned, non-overlapping nucleosomes must be supplanted by a more dynamic view in which nucleosomes, aided by chromatin remodelers, transiently assemble and disassemble, translocate, and interact with each other and with other chromatin components such as regulatory factors and transcriptional machinery.