The Internet of Things (IoT) promises ubiquitous connectivity of everything everywhere, which represents the biggest technology trend in the years to come. It is expected that by 2020 over 25 billion devices will be connected to cellular networks; far beyond the number of devices in current wireless networks. Machine-to-Machine (M2M) communications aims at providing the communication infrastructure for enabling IoT by facilitating the billions of multi-role devices to communicate with each other and with the underlying data transport infrastructure without, or with little, human intervention. Providing this infrastructure will require a dramatic shift from the current protocols mostly designed for human-to-human (H2H) applications. This article reviews recent 3GPP solutions for enabling massive cellular IoT and investigates the random access strategies for M2M communications, which shows that cellular networks must evolve to handle the new ways in which devices will connect and communicate with the system. A massive non-orthogonal multiple access (NOMA) technique is then presented as a promising solution to support a massive number of IoT devices in cellular networks, where we also identify its practical challenges and future research directions.