Advances in artificial intelligence (AI), especially deep learning (DL), have facilitated magnetic resonance imaging (MRI) data analysis, enabling AI-assisted medical image diagnoses and prognoses. However, most of the DL models are considered as “black boxes”. There is an unmet need to demystify DL models so domain experts can trust these high-performance DL models. This has resulted in a sub-domain of AI research called explainable artificial intelligence (XAI). In the last decade, many experts have dedicated their efforts to developing novel XAI methods that are competent at visualizing and explaining the logic behind data-driven DL models. However, XAI techniques are still in their infancy for medical MRI image analysis. This study aims to outline the XAI applications that are able to interpret DL models for MRI data analysis. We first introduce several common MRI data modalities. Then, a brief history of DL models is discussed. Next, we highlight XAI frameworks and elaborate on the principles of multiple popular XAI methods. Moreover, studies on XAI applications in MRI image analysis are reviewed across the tissues/organs of the human body. A quantitative analysis is conducted to reveal the insights of MRI researchers on these XAI techniques. Finally, evaluations of XAI methods are discussed. This survey presents recent advances in the XAI domain for explaining the DL models that have been utilized in MRI applications.