Money laundering has been a global issue for decades, which is one of the major threat for economy and society. Government, regulatory and financial institutions are combating it together in their respective capacity, however still billions of dollars in fines by authorities make the headlines in the news. High-speed internet services have enabled financial institutions to deliver better customer experience through multi-channel engagements, which has led to exponential growth in transactions and new avenues for laundering the money for fraudsters. Literature shows the usage of statistical methods, data mining and Machine Learning (ML) techniques for money laundering detection, but limited research on Deep Learning (DL) techniques, primarily due to lack of model interpretability and explainability of the decisions made. Several studies are conducted on application of ML for Anti-Money Laundering (AML), and Explainable Artificial Intelligence (XAI) techniques in general, but lacks the study on usage of DL techniques together with XAI. This paper aims to review the current state-of-the-art literature on DL together with XAI for identifying suspicious money laundering transactions and identify future research areas. Key findings of the review are, researchers have preferred variants of Convolutional Neural Networks, and AutoEncoder; graph deep learning together with natural language processing is emerging as an important technology for AML; XAI use is not seen in AML domain; 51% ML methods used in AML are non-interpretable, 58% studies used sample of old real data; key challenges for researchers are access to recent real transaction data and scarcity of labelled training data; and data being highly imbalanced. Future research directions are, application of XAI techniques to bring-out explainability, graph deep learning using natural language processing (NLP), unsupervised and reinforcement learning to handle lack of labelled data; and joint research programs between research community and industry to benefit from domain knowledge and controlled access to data.