Abstract Conventional neural networks are software simulations of artificial neural networks (ANNs) implemented on von Neumann machines. This technology has recently encountered bottlenecks in terms of computing speed and energy consumption, leading to increased research interest in optical neural networks (ONNs), which are expected to become the basis for the next generation of artificial intelligence. To provide a better understanding of ONNs and to motivate further developments in this field, previous studies of ONN are reviewed in this article. Our work mainly focuses on the mathematical operations that are decomposed from theoretical models of ANNs and their corresponding optical implementations; these include matrix multiplication, nonlinear activation, convolution, and learning algorithms realized via optical approaches. Some fundamental information about ANNs is also introduced to make this work friendlier to non-experts.