The Multiple Access Channel (MAC) is one of the most well studied and understood network information theoretic models, describing a scenario where K users wish to deliver their information message to one receiver, sharing the same transmission channel. Beyond the multitude of practical and somewhat heuristic MAC protocols (e.g., TDMA, FDMA, CDMA, CSMA, Aloha, and variations thereof), the information theoretic capacity region is well understood under many situations of interest, and in particular in the Gaussian case, modeling the uplink of a wireless system with one access point or base station (receiver) and several users, sharing the same frequency band. \\ \\ More recently, a variant of this model has been proposed for a situation where a very large (virtually unlimited) number of users wish to communicate only very sporadically, such that at any point in time only a finite and relatively small number of users are ``active''. This scenario is appropriate for machine-type communications and Internet of Things, where a multitude of sensors and objects have only rather sporadic data to send, but they need to send them when they are created, at random times. \\ \\ The identification of the active user set, or the active message set (the list of messages transmitted, irrespectively of who is transmitting them) has some points in common with a compressed sensing problem, where the activity vector (entry 1 if a user/message is active and 0 otherwise) is the key object to be estimated at the receiver side. \\ \\ In this talk we review the basic MAC model and results, a variant for massive random access called ``unsourced random access'' where all users use the same codebook, and related recent results and algorithms. including some capacity scaling for this model, which remains quite open as far as a full information theoretic characterization is concerned.