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Generalized Nash equilibrium problems (GNEPs) are a generalization of the classic Nash equilibrium problems (NEPs), where each player's strategy set depends on the choices of the other players. In this work we study constraint qualifcations (CQs) and optimality conditions tailored for GNEPs, and we discuss their relations and implications for global convergence of algorithms. We show the surprising fact that, in contrast to the case of nonlinear programming, in general the Karush-Kuhn-Tucker (KKT) residual cannot be made arbitrarily small near a solution of a GNEP. We then discuss some important practical consequences of this fact. We also prove that this phenomenon is not present in an important class of GNEPs, including NEPs. Finally, under an introduced weak CQ, we prove global convergence to a KKT point of an augmented Lagrangian algorithm for GNEPs, and under the quasi-normality (QN) CQ for GNEPs, we prove boundedness of the dual sequence. [ABSTRACT FROM AUTHOR]