Deltas on planet Mars record past climate, but so far a wide range of hypotheses for their formation have been proposed. The objective of this paper is to understand martian fan deltas, their formative conditions, evolution and formative duration, and implications for the past climate.As an introduction to Mars, physiographic provinces are described and unambiguous proof is listed for the presence of fl owing water in the past, such as certain minerals, groundwater, catastrophic outfl ow channels, alluvial fans and fan deltas, distributary networks and glaciers. The climate history of Mars differs from that of Earth by having had much drier conditions than on Earth, extreme intermittency and extreme events, most of them billions of years ago.Tens of fan deltas, unchannelized fan deltas and stepped fans have been found in impact crater and other lakes. The stepped fans were likely formed by backstepping under fast rising lake levels and have no known terrestrial equivalent. The fan deltas were formed once the lake overfl owed. Alluvial fans are much more numerous and formed with less water.The delta studies illustrate how major challenges of martian morphology and reconstruction of past conditions can be taken up most effectively by combinations of the available high-resolution images and digital elevation models, terrestrial analogues, laboratory experiments and physics-based models gleaned from geomorphology.To resolve formative mechanisms and time scale of martian fans and deltas, morphological distinctions between dense debris fl ows and dilute fl uvial fl ows must be identifi ed for both source and sink areas. Furthermore, the properties of the martian surface material are very poorly constrained but can be explored by modelling various mass wasting, fl uvial and glacial phenomena and hydrology, and by experimentation with slightly cohesive sediment. Finally, the highly debated role of groundwater sapping in valley and delta formation on Mars should be explored experimentally.