In this review paper, we discuss the properties and applications of photonic computing and analog signal processing. Photonic computational circuits have large operation bandwidth, low power consumption, and fine frequency control, enabling a wide range of application-specific computational techniques that are impossible to implement using traditional electrical and digital hardware alone. These advantages are illustrated in the elegant implementation of optical steganography, the real-time blind separation of signals in the same bandwidth, and the efficient acceleration of artificial neural network inference. The working principles and use of photonic circuits for analog signal processing and neuromorphic computing are reviewed and notable demonstrated applications are highlighted.