The simultaneous advances in artificial neural networks and photonic integration technologies have spurred extensive research in optical computing and optical neural networks (ONNs). The potential to simultaneously exploit multiple physical dimensions of time, wavelength and space give ONNs the ability to achieve computing operations with high parallelism and large-data throughput. Different photonic multiplexing techniques based on these multiple degrees of freedom have enabled ONNs with large-scale interconnectivity and linear computing functions. Here, we review the recent advances of ONNs based on different approaches to photonic multiplexing, and present our outlook on key technologies needed to further advance these photonic multiplexing/hybrid-multiplexing techniques of ONNs.