A challenging problem in the protection of power transformers is the fault detection and diagnosis (FDD). FDD has an essential role in the reliability and safety of modern power systems; thus, it has been recently the center of attention in both industrial and academic studies. Due to unpredictable nature of fault, it should be located and isolated fast so that its impact on transformers is minimized. The main advantage of FDD is that it prevents costly repairs, costly downtimes, putting human into danger, and destruction of the equipment nearby. Thus, understanding failure modes, their cause and effects, and developing real-time automated devices for fault diagnosis with the ability to capture the early fault signs. Recently, various studies have been conducted on FDD in transformers using different views, methods, constraints, and objectives. There are good reviews in this context, but they are mainly focused on a specific area of this vast context. The purpose of this study is to classify the publications and make a systematic review of the FDD techniques and algorithms from different aspects and views from 1990 to 2020. This paper also summarizes the pros and cons of the existing methods. This paper provides a comprehensive background for future studies by evaluating the studies of this area and categorizing them.