Abstract. We study equilibrium models governed by parameter-dependent quasivariational inequalities important from the viewpoint of optimization/equilibrium theory as well as numerous applications. The main attention is paid to quasivariational inequalities with parameters entering both single-valued and multivalued parts of the corresponding generalized equations in the sense of Robinson. The main tools of our variational analysis involve coderivatives of solution maps to quasivariational inequalities, which allow us to obtain efficient conditions for robust Lipschitzian stability of quasivariational inequalities and also to derive new necessary optimality conditions for mathematical programs with quasivariational constraints. To conduct this analysis, we develop new results on coderivative calculus for structural settings involved in our models. The results obtained are illustrated by applications to some optimization and equilibrium models related to parameterized Nash games of two players and to oligopolistic market equilibria.