SAR Automatic Target Recognition (ATR) is a key task in microwave remote sensing. Recently, Deep Neural Networks (DNNs) have shown promising results in SAR ATR. However, despite the success of DNNs, their underlying reasoning and decision mechanisms operate essentially like a black box and are unknown to users. This lack of transparency and explainability in SAR ATR pose a severe security risk and reduce the users’ trust in and the verifiability of the decision-making process. To address these challenges, in this paper, we argue that research on the explainability and interpretability of SAR ATR is necessary to enable development of interpretable SAR ATR models and algorithms, and thereby, improve the validity and transparency of AI-based SAR ATR systems. First, we present recent developments in SAR ATR, note current practical challenges, and make a plea for research to improve the explainability and interpretability of SAR ATR. Second, we review and summarize recent research in and practical applications of explainable machine learning and deep learning. Further, we discuss aspects of explainable SAR ATR with respect to model understanding, model diagnosis, and model improvement toward a better understanding of the internal representations and decision mechanisms. Moreover, we emphasize the need to exploit interpretable SAR feature learning and recognition models that integrate SAR physical characteristics and domain knowledge. Finally, we draw our conclusion and suggest future work for SAR ATR that combines data and knowledge-driven methods, human–computer cooperation, and interactive deep learning.