Stress granules (SGs) and processing bodies (PBs) are membraneless ribonucleoprotein-based cellular compartments that assemble in response to stress. SGs and PBs form through liquid–liquid phase separation that is driven by high local concentrations of key proteins and RNAs, both of which dynamically shuttle between the granules and the cytoplasm. SGs uniquely contain certain translation initiation factors and PBs are uniquely enriched with factors related to mRNA degradation and decay, although recent analyses reveal much broader protein commonality between these granules. Despite detailed knowledge of their composition and dynamics, the function of SGs and PBs remains poorly understood. Both, however, contain mRNAs, implicating their assembly in the regulation of RNA metabolism. SGs may also serve as hubs that rewire signaling events during stress. By contrast, PBs may constitute RNA storage centers, independent of mRNA decay. The aberrant assembly or disassembly of these granules has pathological implications in cancer, viral infection and neurodegeneration. Here, we review the current concepts regarding the formation, composition, dynamics, function and involvement in disease of SGs and PBs.