The next wave of wireless technologies will proliferate in connecting sensors, machines, and robots for myriad new applications, thereby creating the fabric for the Internet of Things (IoT). A generic scenario for IoT connectivity involves a massive number of machine-type connections. But in a typical application, only a small (unknown) subset of devices are active at any given instant, thus one of the key challenges for providing massive IoT connectivity is to detect the active devices first and then to decode their data with low latency. This article outlines several key signal processing techniques that are applicable to the problem of massive IoT access, focusing primarily on advanced compressed sensing technique and its application for efficient detection of the active devices. We show that massive multiple-input multiple-output (MIMO) is especially well-suited for massive IoT connectivity in the sense that the device detection error can be driven to zero asymptotically in the limit as the number of antennas at the base station goes to infinity by using the multiple-measurement vector (MMV) compressed sensing techniques. The paper also provides a perspective on several related important techniques for massive access, such as embedding of short messages onto the device activity detection process and the coded random access.