As we enter the post-Moore era, heterogeneous optoelectronic integrated circuits (OEICs) are attracting significant attention as an alternative approach to scaling to smaller-sized transistors. Two-dimensional (2D) materials, offering a range of intriguing optoelectronic properties as semiconductors, semimetals, and insulators, provide great potential for developing nextgeneration heterogeneous OEICs. For instance, Fermi levels of 2D materials can be tuned by applying electrical voltages, while their atomically thin geometries are inherently suited for the fabrication of planar devices without suffering from lattice mismatch. Since the first graphene-on-silicon OEICs were demonstrated in 2011, 2D-material heterogeneous OEICs have significantly progressed. To date, researchers have a better understanding of the importance of interface states on the optical properties of chip-integrated 2D materials. Moreover, there has been impressive progress towards the use of 2D materials for waveguide-integrated lasers, modulators, and photodetectors. In this review, we summarize the history, status, and trend of integrated optoelectronics with 2D materials.