Nowadays, the main grid is facing several challenges related to the integration of renewable energy resources, deployment of grid-level energy storage devices, deployment of new usages such as the electric vehicle, massive usage of power electronic devices at different electric grid stages and the inter-connection with microgrids and prosumers. To deal with these challenges, the concept of a smart, fault-tolerant, and self-healing power grid has emerged in the last few decades to move towards a more resilient and efficient global electrical network. The smart grid concept implies a bi-directional flow of power and information between all key energy players and requires smart information technologies, smart sensors, and low-latency communication devices. Moreover, with the increasing constraints, the power grid is subjected to several disturbances, which can evolve to a fault and, in some rare circumstances, to catastrophic failure. These disturbances include wiring issues, grounding, switching transients, load variations, and harmonics generation. These aspects justify the need for real-time condition monitoring of the power grid and its subsystems and the implementation of predictive maintenance tools. Hence, researchers in industry and academia are developing and implementing power systems monitoring approaches allowing pervasive and effective communication, fault diagnosis, disturbance classification and root cause identification. Specifically, a focus is placed on power quality monitoring using advanced signal processing and machine learning approaches for disturbances characterization. Even though this review paper is not exhaustive, it can be considered as a valuable guide for researchers and engineers who are interested in signal processing approaches and machine learning techniques for power system monitoring and grid-disturbance classification purposes.