The rise of artificial neural networks and silicon photonics has reignited interest in neuromorphic photonics, the emulation of neural systems in photonic hardware. Here, we summarize past work in neuromorphic photonics, with emphasis on silicon photonics implementations. Suitable neuronal models are first discussed, with separate discussions on implementation of linear weighted summing and nonlinear activation. Networking techniques to achieve neural networks proper is then discussed. Next, we review three applications of neuromorphic photonics: ODE solving, model-predictive control, and intelligent signal processing. We finish with a short outlook.