We study the surfing motion of chemically-active objects along a flat liquid-gas interface. The objects are self-propelled by means of creating and maintaining a surface tension gradient resulting from an asymmetric release of a chemical agent. We use theory, numerical simulation, and experiments to examine the propulsion of these Marangoni surfers. First, we consider the behavior of simple surfers at a wide range of release rates and diffusivity of the exuded chemical species (that set the effective Reynolds and Peclet numbers), and subject to various degrees of confinement represented by the thickness of the liquid film. We show that the surfers can undergo a forward, backward, or an arrested motion, and identify the link between these modes of mobility and the flow structure in the vicinity of the surfers. We also find, perhaps surprisingly, that there exist certain intermediate Reynolds and Peclet numbers at which the propulsion speed of the surfers reaches a peak. Next, we focus on the dynamics of a self-powered and remotely controlled robot, which is inspired by water-walking insects and uses the Marangoni effect for both forward motion and change of direction. We analyze the performance of this surfing robot (characterized by its speed, fuel efficiency, and maneuverability) and showcase its novel capabilities. Overall, the findings of our investigations provide new insights into the propulsion of Marangoni-driven surfers and help establish engineering guidelines for their efficient design and effective operation.