This open access book presents an overview of recent advances in photonic neural networks with spatiotemporal dynamics. The computing and implementation paradigms presented in this book are outcomes of interdisciplinary studies by collaborative researchers from the three fields of nonlinear mathematical science, information photonics, and integrated systems engineering. This book offers novel multidisciplinary viewpoints on photonic neural networks, illustrating recent advances in three types of computing methodologies: fluorescence energy transfer computing, spatial-photonic spin system, and photonic reservoir computing. The book consists of four parts: Part I introduces the backgrounds of optical computing and neural network dynamics; Part II presents fluorescence energy transfer computing, a novel computing technology based on nanoscale networks of fluorescent particles; Parts III and IV review the models and implementation of spatial-photonic spin systems and photonic reservoir computing, respectively. These contents are beneficial to researchers in a broad range of fields, including information science, mathematical science, applied physics, and engineering, to better understand the novel computing concepts of photonic neural networks with spatiotemporal dynamics.