The eukaryotic cell nucleus is one of the most complex cell organelles. Despite the absence of membranes, the nuclear space is divided into numerous compartments where different processes involved in genome activity take place. The most important nuclear compartments include nucleoli, nuclear speckles, PML bodies, Cajal bodies, histone locus bodies, Polycomb bodies, insulator bodies, and transcription and replication factories. A structural basis for the nuclear compartmentalization is provided by genomic DNA, which occupies most of the nuclear volume. In turn, nuclear compartments guide the chromosome folding by providing a platform for spatial interactions of individual genomic loci. The review discusses the fundamental principles of higher-order genome organization with a focus on chromosome territories and chromosome domains and considers the structure and function of key nuclear compartments. It is shown that the functional compartmentalization of the cell nucleus is tightly interconnected with genome spatial organization, is highly dynamic, and is based on stochastic processes.