Metasurfaces are artificially structured thin films with unusual properties on demand. Different from metamaterials, the metasurfaces change the electromagnetic waves mainly by exploiting the boundary conditions, rather than the constitutive parameters in three dimensional (3D) spaces. Despite the intrinsic similarities in the operational principles of metasurfaces, there is not a universal theory available for the understanding and design of these devices. In this article, we propose the concept of metasurface waves (M-waves) and provide a general theory to describe the principles of such waves. Most importantly, it is shown that the M-waves share some fundamental properties such as extremely short wavelength, abrupt phase change and strong chromatic dispersion, which making them different from traditional bulk waves. We show that these properties can enable many important applications such as subwavelength imaging and lithography, planar optical devices, broadband anti-reflection, absorption and polarization conversion. Our results demonstrated unambiguously that traditional laws of diffraction, refraction, reflection and absorption can be overcome by using the novel properties of M-waves. The theory provided here may pave the way for the design of new electromagnetic devices and further improvement of metasurfaces.