Massive machine-type communications (mMTC) is one of the main three focus areas in the 5th generation (5G) of mobile standards to enable connectivity of a massive number of internet of things (IoT) devices with little or no human intervention. In conventional human-type communications (HTC), due to the limited number of available radio resources and orthogonal/non-overlapping nature of existing resource allocation techniques, users need to compete for connectivity through a random access (RA) process, which may turn into a performance bottleneck in mMTC. In this context, non-orthogonal multiple access (NOMA) has emerged as a potential technology that allows overlapping of multiple users over a radio resource, thereby creating an opportunity to enable more autonomous and grant-free communication, where devices can transmit data whenever they need. The existing literature on NOMA schemes majorly considers centralized scheduling based HTC, where users are already connected, and various system parameters like spreading sequences, interleaving patterns, power control, etc., are predefined. Contrary to HTC, mMTC traffic is different with mostly uplink communication, small data size per device, diverse quality of service, autonomous nature, and massive number of devices. Hence, the signaling overhead and latency of centralized scheduling becomes a potential performance bottleneck. To tackle this, grant-free access is needed, where mMTC devices can autonomously transmit their data over randomly chosen radio resources. This article, in contrast to existing surveys, comprehensively discusses the recent advances in NOMA from a grant-free connectivity perspective. Moreover, related practical challenges and future directions are discussed.