Reservoir computing offers a versatile approach to temporal information processing by exploiting the dynamics of nonlinear systems to enable learning with a linear readout. Photonic hardware is particularly attractive for this paradigm, thanks to ultrafast dynamics and massive parallelism across multiple accessible degrees of freedom of light. In this thematic review, we chart the development of photonic reservoir computing across hardware platforms, from delay-based architectures and integrated photonic circuit implementations to free-space systems. We also discuss how the input and output layers influence the performance of photonic reservoirs. We present the explored learning algorithms that go beyond linear regression, and list some popular benchmark tasks and shed light on some key applications that have been investigated in the literature. Finally, we discuss emerging unconventional implementations. The overall goal of this review is to provide a panoramic overview of photonic reservoir computing, useful for both practitioners and newcomers to the field.