Search for Unidentified Maritime Objects (SUMO) is an algorithm for ship detection in satellite Synthetic Aperture Radar (SAR) images. It has been developed over the course of more than 15 years, using a large amount of SAR images from almost all available SAR satellites operating in L-, C-and X-band. As validated by benchmark tests, it performs very well on a wide range of SAR image modes (from Spotlight to ScanSAR) and resolutions (from 1-100 m) and for all types and sizes of ships, within the physical limits imposed by the radar imaging. This paper describes, in detail, the algorithmic approach in all of the steps of the ship detection: land masking, clutter estimation, detection thresholding, target clustering, ship attribute estimation and false alarm suppression. SUMO is a pixel-based CFAR (Constant False Alarm Rate) detector for multi-look radar images. It assumes a K distribution for the sea clutter, corrected however for deviations of the actual sea clutter from this distribution, implementing a fast and robust method for the clutter background estimation. The clustering of detected pixels into targets (ships) uses several thresholds to deal with the typically irregular distribution of the radar backscatter over a ship. In a multi-polarization image, the different channels are fused. Azimuth ambiguities, a common source of false alarms in ship detection, are removed. A reliability indicator is computed for each target. In post-processing, using the results of a series of images, additional false alarms from recurrent (fixed) targets including range ambiguities are also removed. SUMO can run in semi-automatic mode, where an operator can verify each detected target. It can also run in fully automatic mode, where batches of over 10,000 images have successfully been processed in less than two hours. The number of satellite SAR systems keeps increasing, as does their application to maritime surveillance. The open data policy of the EU's Copernicus program, which includes the Sentinel-1 satellite, has hugely increased the availability of SAR images. This paper aims to cater to the consequently expected wider demand for knowledge about SAR ship detectors.