Automatic modulation recognition (AMR) detects the modulation scheme of the received signals for further signal processing without needing prior information, and provides the essential function when such information is missing. Recent breakthroughs in deep learning (DL) have laid the foundation for developing high-performance DL-AMR approaches for communications systems. Comparing with traditional modulation detection methods, DL-AMR approaches have achieved promising performance including high recognition accuracy and low false alarms due to the strong feature extraction and classification abilities of deep neural networks. Despite the promising potential, DL-AMR approaches also bring concerns to complexity and explainability, which affect the practical deployment in wireless communications systems. This paper aims to present a review of the current DL-AMR research, with a focus on appropriate DL models and benchmark datasets. We further provide comprehensive experiments to compare the state of the art models for single-input-single-output (SISO) systems from both accuracy and complexity perspectives, and propose to apply DL-AMR in the new multiple-input-multiple-output (MIMO) scenario with precoding. Finally, existing challenges and possible future research directions are discussed.