Ultrasound (US) has become one of the most commonly performed imaging modalities in clinical practice. It is a rapidly evolving technology with certain advantages and with unique challenges that include low imaging quality and high variability. From the perspective of image analysis, it is essential to develop advanced automatic US image analysis methods to assist in US diagnosis and/or to make such assessment more objective and accurate. Deep learning has recently emerged as the leading machine learning tool in various research fields, and especially in general imaging analysis and computer vision. Deep learning also shows huge potential for various automatic US image analysis tasks. This review first briefly introduces several popular deep learning architectures, and then summarizes and thoroughly discusses their applications in various specific tasks in US image analysis, such as classification, detection, and segmentation. Finally, the open challenges and potential trends of the future application of deep learning in medical US image analysis are discussed.