Heat Pumps (HPs) market have performed a progressive growth in the last years and the predictions still point to an increase greater than 200% until 2030. Besides the fact that such technology presents a more efficient solution for both heating and cooling, it also does not depend on fossil fuels when electricity is generated from renewable sources. In this vein, HPs reveal nowadays of a greater importance than ever, and to ensure their availability and reliability a proper Fault Detection and Diagnostics (FDD) of their operation is crucial. This work focuses on an extensive review of the most common faults that may occur in HPs, and the developed work to diagnose each one of them along the last 30 years. Such work relies in different approaches, from rules-based methods to deep learning ones, for each one of the most common faults. Clearly, the most investigated type of fault is the refrigerant undercharge. That is due to the fact that in addition to being the most frequent fault and resulting in a considerable degradation in system's performance, refrigerant undercharge potentially indicates refrigerant leakage, what may result in great impacts in the environment, depending on the type of refrigerant employed.