Massive access, also known as massive connectivity or massive machine-type communication (mMTC), is one of the main use cases of the fifth-generation (5G) and beyond 5G (B5G) wireless networks. A typical application of massive access is the cellular Internet of Things (IoT). Different from conventional human-type communication, massive access aims at realizing efficient and reliable communications for a massive number of IoT devices. Hence, the main characteristics of massive access include low power, massive connectivity, and broad coverage, which require new concepts, theories, and paradigms for the design of next-generation cellular networks. This paper presents a comprehensive survey of massive access design for B5G wireless networks. Specifically, we provide a detailed review of massive access from the perspectives of theory, protocols, techniques, coverage, energy, and security. Furthermore, several future research directions and challenges are identified.