Modern day wireless communication systems rely heavily on the random access schemes that were originally developed in the 1970s. The pioneering Aloha protocol has become a key component of many communications standards, ranging from satellite networks to ad hoc and cellular scenarios. Recent developments, however, have seen a fundamental leap forward based on using the principles of successive interference cancellation. The modern random access protocols using these new techniques have opened up a wealth of new applications. This tutorial style monograph explores the main ideas and design principles that are behind some of these novel schemes. It introduces the reader to the analytical tools used to model such performance. Focussing on slotted solutions it shows how Aloha can be combined with successive interference cancellation and optimized using the theory of codes on graphs. These techniques are illustrated in applications using physical layer network coding to resolve collisions among users and receiver diversity. It concludes by introducing the reader to the recently asynchronous (or unslotted) schemes. Modern Random Access Protocols is a comprehensive and erudite introduction to a set of techniques that will form part of many future modern communication or networked system.