The transverse spatial effects observed in photon pairs produced by parametric down-conversion provide a robust and fertile testing ground for studies of quantum mechanics, non-classical states of light, correlated imaging and quantum information. Over the last 20 years there has been much progress in this area, ranging from technical advances and applications such as quantum imaging to investigations of fundamental aspects of quantum physics such as complementarity relations, Bell's inequality violation and entanglement. The field has grown immensely: a quick search shows that there are hundreds of papers published in this field. The objective of this article is to review the building blocks and major theoretical and experimental advances in the field, along with some possible technical applications and connections to other research areas.

116 pages, 35 figures. To appear in Physics Reports