Machine and deep learning have proven their utility to generate data-driven models with high accuracy and precision. However, their non-linear, complex structures are often difficult to interpret. Consequently, many scholars have developed a plethora of methods to explain their functioning and the logic of their inferences. This systematic review aimed to organise these methods into a hierarchical classification system that builds upon and extends existing taxonomies by adding a significant dimension—the output formats. The reviewed scientific papers were retrieved by conducting an initial search on Google Scholar with the keywords “explainable artificial intelligence”; “explainable machine learning”; and “interpretable machine learning”. A subsequent iterative search was carried out by checking the bibliography of these articles. The addition of the dimension of the explanation format makes the proposed classification system a practical tool for scholars, supporting them to select the most suitable type of explanation format for the problem at hand. Given the wide variety of challenges faced by researchers, the existing XAI methods provide several solutions to meet the requirements that differ considerably between the users, problems and application fields of artificial intelligence (AI). The task of identifying the most appropriate explanation can be daunting, thus the need for a classification system that helps with the selection of methods. This work concludes by critically identifying the limitations of the formats of explanations and by providing recommendations and possible future research directions on how to build a more generally applicable XAI method. Future work should be flexible enough to meet the many requirements posed by the widespread use of AI in several fields, and the new regulations.