Layered materials are very attractive for studies of light-matter interactions at the nanoscale. In particular, isolated quantum systems such as color centers and quantum dots embedded in these materials are gaining interest due to their potential use in a variety of quantum technologies and nanophotonics. Here, we review the field of nonclassical light emission from van der Waals crystals and atomically thin two-dimensional materials. We focus on transition metal dichalcogenides and hexagonal boron nitride and discuss the fabrication and properties of quantum emitters in these systems and proof-of-concept experiments that provide a foundation for their integration in on-chip nanophotonic circuits. These experiments include tuning of the emission wavelength, electrical excitation, and coupling of the emitters to waveguides, dielectric cavities, and plasmonic resonators. Finally, we discuss current challenges in the field and provide an outlook to further stimulate scientific discussion.