Photonics offers exciting opportunities for neuromorphic computing. This paper specifically reviews the prospects of integrated optical solutions for accelerating inference and training of artificial neural networks. Calculating the synaptic function, thereof, is computationally very expensive and does not scale well on state-of-the-art computing platforms. Analog signal processing, using linear and nonlinear properties of integrated optical devices, offers a path toward substantially improving performance and power efficiency of these artificial intelligence workloads. The ability of integrated photonics to operate at very high speeds opens opportunities for time-critical real-time applications, while chip-level integration paves the way to cost-effective manufacturing and assembly.