Random access represents possibly the simplest and yet one of the best known approaches for sharing a channel among several users. Since their introduction in the 1970s, random access schemes have been thoroughly studied and small variations of the pioneering Aloha protocol have since then become a key component of many communications standards, ranging from satellite networks to ad hoc and cellular scenarios. A fundamental step forward for this old paradigm has been witnessed in the past few years, with the development of new solutions, mainly based on the principles of successive interference cancellation, which made it possible to embrace constructively collisions among packets rather enduring them as a waste of resources. These new lines of research have rendered the performance of modern random access protocols competitive to that of their coordinated counterparts, paving the road for a multitude of new applications. This monograph explores the main ideas and design principles that are behind some of such novel schemes, and aims at offering to the reader an introduction to the analytical tools that can be used to model their performance. After reviewing some relevant results for the random access channel, the volume focuses on slotted solutions that combine the approach of diversity Aloha with successive interference cancellation, and discusses their optimisation based on an analogy with the theory of codes on graphs. The potential of modern random access is then further explored considering two families of schemes: the former based on physical layer network coding to resolve collisions among users, and the latter leaning on the concept of receiver diversity. Finally, the opportunities and the challenges encountered by random access solutions recently devised to operate in asynchronous, i.e., unslotted, scenarios are reviewed and discussed.