Critical phenomena at surfaces of a semi-infinite O(n) spin model are reviewed from the viewpoint of several theoretical approaches including the scaling theory, the high-temperature expansion for arbitrary number n, the field theoretical ε (= 4 - d) expansion also for arbitrary n, and the 1/n expansion for arbitrary spatial dimension d. How the mathematical complication in the analysis of a semi-infinite system due to the lack of the translational invariance can be overcome in the ε and 1/n expansion is described in detail. The values for the surface critical exponents and some other universal quantities are obtained for several types of phase transition, i.e. the ordinary, special, anisotropic-special and extraordinary transitions. The form of the two-spin correlation function at criticality is discussed explicitly by means of the ε and 1/n expansions; it is found that the correlation function depends on the real and image distances only, which is a consequence of the conformal invariance. Relation to the problem on surface-grafted polymer networks in dilute solution is mentioned also.