Chromatin is partitioned on multiple length scales into subcompartments that differ from each other with respect to their molecular composition and biological function. It is a key question how these compartments can form even though diffusion constantly mixes the nuclear interior and rapidly balances concentration gradients of soluble nuclear components. Different biophysical concepts are currently used to explain the formation of "chromatin bodies" in a self-organizing manner and without consuming energy. They rationalize how soluble protein factors that are dissolved in the liquid nuclear phase, the nucleoplasm, bind and organize transcriptionally active or silenced chromatin domains. In addition to cooperative binding of proteins to a preformed chromatin structure, two different mechanisms for the formation of phase-separated chromatin subcompartments have been proposed. One is based on bridging proteins that cross-link polymer segments with particular properties. Bridging can induce a collapse of the nucleosome chain and associated factors into an ordered globular phase. The other mechanism is based on multivalent interactions among soluble molecules that bind to chromatin. These interactions can induce liquid-liquid phase separation, which drives the assembly of liquid-like nuclear bodies around the respective binding sites on chromatin. Both phase separation mechanisms can explain that chromatin bodies are dynamic spherical structures, which can coalesce and are in constant and rapid exchange with the surrounding nucleoplasm. However, they make distinct predictions about how the size, density, and stability of chromatin bodies depends on the concentration and interaction behavior of the molecules involved. Here, we compare the different biophysical mechanisms for the assembly of chromatin bodies and discuss experimental strategies to distinguish them from each other. Furthermore, we outline the implications for the establishment and memory of functional chromatin state patterns.