The notion of generalized equation, that is an inclusion involving a single-valued function f and a normal cone mapping N(· ; C) to a constraint set C in the form 0 ∈ f(x) + N(x; C), was introduced by Stephen M. Robinson in 1979. Generalized equations are convenient tools for formulating problems in complementarity, mathematical programming, and variational inequalities. Generalized differentiability properties of normal cone mappings allow one to obtain useful information about solution stability of parametric generalized equations as well as solution stability of problems that refer to parametric generalized equations. In this presentation, we give formulas for computing the Fréchet and Mordukhovich coderivatives of normal cone mappings to constraint sets with respect to perturbed polyhedral convex sets, perturbed Euclidean balls, perturbed smooth-boundary sets. The obtained coderivative formulas are applied to establish necessary and sufficient conditions for the local Lipschitz-like property and metric regularity of the solution maps of parametric generalized equations as well as the solution maps of parametric variational inequalities, the Karush-Kuhn-Tucker point set maps of trust-region subproblems in trust-region methods.