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This paper considers generalized convexity and concavity properties of the optimal value function f* for the general parametric optimization problem P(e) of the form min x sub f (x,e) s.t. x epsilon R(e). Many results on convexity and concavity characterizations of f* were presented by the authors in a previous paper. Such properties of f* and the solution map S* form an important part of the theoretical basis for sensitivity, stability and parametric analysis in mathematical optimization. The authors give sufficient conditions for several types of generalized convexity and concavity of f*, in terms of respective generalized convexity and concavity assumptions on f and convexity and concavity assumptions on 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. (Author)