The goal of integrated quantum photonics is to combine components for the generation, manipulation, and detection of nonclassical light in a phase-stable and efficient platform. Solid-state quantum emitters have recently reached outstanding performance as single-photon sources. In parallel, photonic integrated circuits have been advanced to the point that thousands of components can be controlled on a chip with high efficiency and phase stability. Consequently, researchers are now beginning to combine these leading quantum emitters and photonic integrated circuit platforms to realize the best properties of each technology. In this paper, we review recent advances in integrated quantum photonics based on such hybrid systems. Although hybrid integration solves many limitations of individual platforms, it also introduces new challenges that arise from interfacing different materials. We review various issues in solid-state quantum emitters and photonic integrated circuits, the hybrid integration techniques that bridge these two systems, and methods for chip-based manipulation of photons and emitters. Finally, we discuss the remaining challenges and future prospects of on-chip quantum photonics with integrated quantum emitters. (C) 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement