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Swimming droplets are artificial microswimmers based on liquid droplets that show self-propelled motion when immersed in a second liquid. These systems are of tremendous interest as experimental models for the study of collective dynamics far from thermal equilibrium. For biological systems, such as bacterial colonies, plankton, or fish swarms, swimming droplets can provide a vital link between simulations and real life. We review the experimental systems and discuss the mechanisms of self-propulsion. Most systems are based on surfactant-stabilized droplets, the surfactant layer of which is modified in a way that leads to a steady Marangoni stress resulting in an autonomous motion of the droplet. The modification of the surfactant layer is caused either by the advection of a chemical reactant or by a solubilization process. Some types of swimming droplets possess a very simple design and long active periods …