We consider 2D surface superconductivity in high magnetic fields parallel to the surface. We demonstrate that the spin-orbit interaction at the surface changes the properties of the inhomogeneous superconducting Larkin-Ovchinnikov-Fulde-Ferrell state that develops above fields given by the paramagnetic criterion. Strong spin-orbit interaction significantly broadens the range of existence of the LOFF phase, which takes the form of periodic superconducting stripes running along the field direction on the surface, leading to the anisotropy of its properties. In connection with experiments by J.H. Schön et al. [Nature 914, 434 (2001)] on superconductivity of electrically doped films of the cuprate material CaCuO2, we also discuss this problem for the d-wave pairing to indicate the possibility of a re-orientation transition as the magnetic field direction is rotated in the plane parallel to the surface. Our results provide a tool for studying surface superconductivity as a function of doping. Islands of a surface superconducting phase were also observed for the surface-doped tungsten bronzes, WO 3 :Na, at T c = 91.5K [4] . The ARPES results mentioned above suggest that SSC may actually be a rather widespread phenomenon.It is noteworthy that the bulk WO 3 is an insulator at low doping [5]. Recently a remarkable breakthrough has been accomplished by doping insulator surfaces electrically in the so-called Field-Effect Transistor (FET) geometry [6]. SSC with high T c was induced by both electronic and hole doping of films of the prototype cuprate material, CaCuO 2 [7].All this makes us believe that the search for SSC emerges as a new and important development in studies of the properties of surfaces, especially their metallic properties. As for the SSC itself, its mechanisms are unknown, and may have nothing to do with the ones in the bulk. Of a special challenge is the possibility of superconductivity at surfaces of ordinary metals such as Cu * Also at L.D. Landau Institute for Theoretical Physics, Chernogolovka, 142432, Russia[3] or lithium-doped Mo and W [2], and the influence of substrate on superconductivity in thin films. Thus, it is necessary to return to a more careful investigation of the low temperature properties of surfaces with absorbed atoms and the role of adsorbed atoms as dopants of carriers into the metallic surface zone (e.g., see Ref.[2]) like it was discovered for WO 3 :Na [4].The major challenge to such a program lies in discerning SSC and studying its properties. Even for insulators where doping by FET, ideally, provides an effective control on surface properties, experimental tools to probe SSC are limited in numbers. The Meissner effect due to surface superconducting islands would probably never produce a bulk screening. Thermodynamical probes are also difficult because of the smallness of contributions from surface layers of atomic thicknesses. So far, e.g., in Ref. [6], surface superconductivity has only been detected by measuring resistivity dependence on temperature in the perpendicular-to-pl...