"The purpose of this work is to study the optical properties of surfaces by SHG. In particular, we analyze the atomic structure of the surface, the surface symmetry, the main microscopic contributions to the non-linear processes that converts the fundamental frequency into the SH radiation. We chose silicon, since its surfaces is the most used in technological applications, and study the properties of two of its faces, the (100) and (111). The Si (111) (1times1) is the simplest of all semiconductor surfaces, in the sense that its equilibrium relaxed geometry is almost identical to an ideally terminated (111) surface [67]. Also, as H saturates every Si dangling bond, there are no surface-related electronic states in the forbbiden gap. Experimentally, this surface can be prepared to a high level of quality [67,68]. Indeed, the experiments are performed on a high quality surface produced by a state-of-the-art technique by the german group of professor W. Daum. Also, computationally it is very simple as it has only one atom per unit surface cell. Therefore, this surface presents a very good reference system for SHG studies, since neither the atomic structure nor electronic surface states in the gap region should be of concern when comparing the theoretical and experimental SHG spectra. The microscopic nonlinear surface susceptibility tensor X is evaluated within the polarizable bond model and the microscopic formulation, using both the semi-empirical tight binding (SETB) and the ab-initio schemes. The SHG spectra are compared among the three different approache, and with the available experimental results. We show that SHG is a very powerful tool to study several physical aspects of a semiconductor surface. Based on the previous success, we finally proceed to calculate SHG from the most fascinating of all semiconductor surfaces, the Si (111)(7x7). The reconstruction of this surface involves a unit cell 7 times larger than the ideally terminated (1x1) unit cell, thus implying a tour de force calculation with every surprising results."