Underwater acoustic target recognition (UATR) technology has been implemented widely in the fields of marine biodiversity detection, marine search and rescue, and seabed mapping, providing an essential basis for human marine economic and military activities. With the rapid development of machine-learning-based technology in the acoustics field, these methods receive wide attention and display a potential impact on UATR problems. This paper reviews current UATR methods based on machine learning. We focus mostly, but not solely, on the recognition of target-radiated noise from passive sonar. First, we provide an overview of the underwater acoustic acquisition and recognition process and briefly introduce the classical acoustic signal feature extraction methods. In this paper, recognition methods for UATR are classified based on the machine learning algorithms used as UATR technologies using statistical learning methods, UATR methods based on deep learning models, and transfer learning and data augmentation technologies for UATR. Finally, the challenges of UATR based on the machine learning method are summarized and directions for UATR development in the future are put forward.