Detecting and tracking ships remotely is now required in a wide range of contexts, from military security to illegal immigration control, as well as the management of fisheries and marine protected areas. Among the available methods, radar remote sensing is increasingly used due to its advantages of being rarely affected by cloud cover and allowing image acquisition during both day and night. The growing availability over the past decade of free synthetic aperture radar (SAR) data, such as Sentinel-1 images, enabled the widespread use of C-band images for ship detection. There is, however, a broad range of SAR data processing methods proposed in the literature, challenging the selection of the most appropriate one for a given application. Here, we conducted a systematic review of the literature on ship detection methods using C-band SAR data from 2015 to 2022. The review shows a partition between traditional and deep learning (DL) methods. Earlier methods were mainly based on constant false alarm rate or polarimetry, which require limited computing resources but critically depend on ships' physical environment. Those approaches are gradually replaced by DL, due to the growth of computing capacities, the wide availability of SAR images, and the publication of DL training datasets. However, access to these computing capacities may not be easy for all users, which could become a major obstacle to their development. While both methods have the same objective, they differ both technically and in their approaches to the problem. Traditional methods mainly focus on ship size in spatial units (meters), whereas DL methods are mainly based on the number of ship pixels, regardless of image resolution. These latter methods can result in a lack of information on ship size and, therefore, a lack of knowledge that could be useful to specific applications, such as fisheries and protected area management.