Convexity and concavity properties of the optimal value function \(f^*\) are considered for the general parametric optimization problem P(\(\epsilon)\) of the form \(\min_{x} f(x,\epsilon)\), s.t. \(x\in {\mathbb{R}}(\epsilon)\). Such properties of \(f^*\) and the solution set map \(S^*\) form an important part of the theoretical basis for sensitivity, stability, and parametric analysis in mathematical optimization. Sufficient conditions are given for several standard types of convexity and concavity of \(f^*\), in terms of respective convexity and concavity assumptions on f and the feasible region point-to-set map R. Specializations of these results to the general parametric inequality- equality constrained nonlinear programming problem and its right-hand- side version are provided.