Photonic delay systems have revolutionized the hardware implementation of Recurrent Neural Networks and Reservoir Computing in particular. The fundamental principles of Reservoir Computing strongly benefit a realization in such complex analog systems. Especially delay systems, potentially providing large numbers of degrees of freedom even in simple architectures, can efficiently be exploited for information processing. The numerous demonstrations of their performance led to a revival of photonic Artificial Neural Network. Today, an astonishing variety of physical substrates, implementation techniques as well as network architectures based on this approach have been successfully employed. Important fundamental aspects of analog hardware Artificial Neural Networks have been investigated, and multiple high-performance applications have been demonstrated. Here, we introduce and explain the most relevant aspects of Artificial Neural Networks and delay systems, the seminal experimental demonstrations of Reservoir Computing in photonic delay systems, plus and the most recent and advanced realizations.