Artificial intelligence (AI) advancements, especially deep learning, have significantly improved medical image processing and analysis in various tasks such as disease detection, classification, and anatomical structure segmentation. This work overviews fundamental concepts, state-of-the-art models, and publicly available datasets in the field of medical imaging. First, we introduce the types of learning problems commonly employed in medical image processing and then proceed to present an overview of commonly used deep learning methods, including convolutional neural networks (CNNs), recurrent neural networks (RNNs), and generative adversarial networks (GANs), with a focus on the image analysis task they are solving, including image classification, object detection/localization, segmentation, generation, and registration. Further, we highlight studies conducted in various application areas, encompassing neurology, brain imaging, retinal analysis, pulmonary imaging, digital pathology, breast imaging, cardiac imaging, bone analysis, abdominal imaging, and musculoskeletal imaging. The strengths and limitations of each method are carefully examined, and the paper identifies pertinent challenges that still require attention, such as the limited availability of annotated data, variability in medical images, and the interpretability issues. Finally, we discuss future research directions with a particular focus on developing explainable deep learning methods and integrating multi-modal data.