Many applications in modern technology, such as self-cleaning surfaces and digital microfluidics, require control over individual fluid droplets on flat surfaces. Existing techniques may suffer from side effects resulting from high electric fields and high temperatures. Here, we introduce a markedly different method, termed "mechanowetting," that is based on the surface tension-controlled droplet motion on deforming surfaces. The method is demonstrated by transporting droplets using transverse surface waves on horizontal and (vertically) inclined surfaces at transport velocities equal to the wave speed. We fully capture the fundamental mechanism of the mechanowetting force numerically and theoretically and establish its dependence on the fluid properties, surface energy, and wave parameters. Mechanowetting has the potential to lead to a range of new applications that feature droplet control through dynamic surface deformations.