Photonic accelerators have emerged as promising alternatives to conventional electronic processors since they offer unique advantages such as high parallelism, low propagation loss, and in-propagation computation, making them well-suited for modern machine learning tasks that benefit from scalable parallelism. Fundamental mathematical operations including matrix-vector multiplication, convolution, and nonlinear activation function are readily achieved with all-optical components. Due to the fixed hardware sizes, most kernel reutilization with photonics is based on the intermediate optical-electrical-optical conversion and storage. Yet, for a class of algorithms where signal recurrence is intrinsic, such truncation is suboptimal. The advancement in photonic material platforms with low loss and high integration density make direct optical signal feedback with on-chip waveguides feasible. This development enables a novel class of devices that we term Integrated Photonic Recurrent Processors (IPRPs). IPRPs uniquely accelerate computation by incorporating optical delay memory, and bypasses the conventional single-pass photonic computing overheads. This review explores algorithms with inherent recurrence and their implementation using integrated photonics. It also highlights potential applications for IPRPs and discusses how emerging material technologies may drive their advancement. With ongoing improvements in fabrication, integration, and control, IPRPs hold strong promise as compact, energy-efficient platforms for advancing optical computing.