In the context of a doomed end of the Moore's law, various new types of computing architectures have been emerging, aiming to meet the demands of intractable computation and artificial intelligence. Photonic computing is a competitive candidate, in light of the inherent properties of photons, including high propagation speed, strong robustness, and multiple degrees of freedom to encode information. Also, the progress of integrated photonics continues to provide novel possibilities, apart from boosting the scalability and stability of photonic computing architectures. Moreover, an introduction of quantum technology might open a new chapter for photonic computing, from the view of single photons. In this Perspective, we highlight the unique features and advances of integrated photonic platforms, whose roles in constructing a non-von Neumann computing architecture are also outlined. We show their potential in solving problems beyond the reach of traditional computers and in machine learning and further discuss the conceivable challenges and opportunities.