Nonlinear boundary control problems for a class of semilinear parabolic systems are considered, from the point of view of semigroup theory. The method is based on some recent general results on parabolic evolution equations with nonlinear boundary conditions. Existence of optimal (boundary) controls is proved using the theory of measurable selections and the Cesari property for multifunctions. Three results are presented covering relaxed controls and controls with state constraints. This generalises, in a substantial way, existing results on linear boundary control problems [M.C. Delfour and M. Sorine, Control of Distributed Parameter Systems, Pergamon Press, Oxford, 1983, pp. 87-90], [I. Lasiecka, Appl. Math. Optim., 6 (1980), pp. 287-383], [P. Acquistapace, et al., SIAM J. Control Optim., 29 (1991), pp. 89-118]. The result presented can be further extended to differential inclusions. Two examples are presented for illustration.