With the profound integration of artificial intelligence and photonics technologies, intelligent photonics is evolving into a disruptive technology that looks poised to revolutionize industries and everyday life. The development of intelligent photonics finds applications in diverse fields, including biomedicine, autonomous driving, and virtual and augmented reality. Artificial intelligence (AI) is fueling a new paradigm of photonics research, providing efficient avenues for optimizing photonics design, advancing optical systems and analyzing optical information. Enabled by the maturity of deep learning, silicon-based optoelectronics, optical materials, and quantum information, photonic computing holds great potential to address the challenges faced by Moore's law and the bottlenecks of the von Neumann architecture. Future implementations of photonic computing may meet the demands for high-performance computing in the digital infrastructure of the information era, such as those posed by 5G, big data, cloud computing, and the Internet of Things. In this study, we summarize recent advances in photonic computing, including on-chip integrated optical neural networks based on micro-ring resonators, multimode interferometers, nanobeam resonators, and subwavelength diffractive units and integration of training and computation. We also highlight the progresses in diffractive neural networks enabled by diffractive optical elements and intelligent metasurfaces, as well as the developments in photonic spiking neural networks, reservoir computing, quantum photonic computing, and large-scale optoelectronic computing chips. In terms of computational optics, we review the advances across a broad range of areas, including computational imaging, microscopy, display, fiber-optic sensing, and laser technologies.