Automatic modulation classification (AMC), which aims to blindly identify the modulation type of an incoming signal at the receiver in wireless communication systems, is a fundamental signal processing technique in the physical layer to improve the spectrum utilization efficiency. Motivated by deep learning (DL) high-impact success in many informatics domains, including radio signal processing for communications, numerous recent AMC methods exploiting deep networks have been proposed to overcome the existing drawbacks of traditional approaches. DL is capable of learning the underlying characteristics of radio signals effectively for modulation pattern recognition, which in turn improves the modulation classification performance under the presence of channel impairments. In this work, we first provide the fundamental concepts of various architectures, such as neural networks, recurrent neural networks, long short-term memory, and convolutional neural networks as the necessary background. We then convey a comprehensive study of DL for AMC in wireless communications, where technical analysis is deliberated in the perspective of state-of-the-art deep architectures. Remarkably, several sophisticated structures and advanced designs of convolutional neural networks are investigated for different data types of sequential radio signals, spectrum images, and constellation images to deal with various channel impairments. Finally, we discuss some primary research challenges and potential future directions in the area of DL for modulation classification.