  We conduct a review of the basic definitions and the principal results in the
study of wavelike spacetimes, that is spacetimes whose metric models massless
radiation moving at the speed of light, focusing in particular on those
geometries with parallel rays. In particular, we motivate and connect their
various definitions, outline their coordinate descriptions and present some
classical results in their study in a language more accessible to modern
readers, including the existence of "null coordinates" and the construction of
Penrose limits. We also present a thorough summary of recent work on causality
in pp-waves, and describe progress in addressing an open question in the field
- the Ehlers-Kundt conjecture.
