Several review studies exist in the literature about monitoring, fault detection, and diagnosis of power transformers. However, they are general approaches in terms of power transformers. Some only focus on applying a specific class of techniques, but again, for general power transformers. Other reviews focus on applying different technologies such as fiber optics, thermal cameras, and vibration sensors, but all within the perspective of general power transformers. A significant question remains: among all types of power transformers, which specific techniques should be used, and why are they more adequate? What are the uncertainties that can decrease their precision? What about the balance, in terms of costs, associated with applying a certain technique and the return needed for a particular type of transformer? In this context, this paper is not only a literature review of well-known problems related to power transformers. Here, we do not just center on large power transformers (100 MVA or higher). Still, we describe a case study of a phase-shifting 1400 MVA-400 kV three-phase transformer that currently connects two European countries that began to show signs of abnormal operating conditions in 2012. In this way, the need to detect and identify anomalies in their initial stage of development for a possible preventive maintenance action is more than justified, which is essentially achieved with continuous monitoring models of the transformer, as concluded in this paper.