A loss of integrity and the effects of damage on mechanical attributes result in macro/micro-mechanical failure, especially in composite structures. As a progressive degradation of material continuity, predictions for any aspects of the initiation and propagation of damage need to be identified by a trustworthy mechanism to guarantee the safety of structures. Besides material design, structural integrity and health need to be monitored carefully. Among the most powerful methods for the detection of damage are machine learning (ML) and deep learning (DL). In this paper, we review state-of-the-art ML methods and their applications in detecting and predicting material damage, concentrating on composite materials. The more influential ML methods are identified based on their performance, and research gaps and future trends are discussed. Based on our findings, DL followed by ensemble-based techniques has the highest application and robustness in the field of damage diagnosis.