The authors introduce a suitable notion of convergence for games, called \({\mathcal N}\)-convergence. This convergence ensures that if each game \(J_ h\) has a Nash solution \(u_ h\), \(J_ h\to^{{\mathcal N}}J_ 0\) and \(u_ h\to u_ 0\), then \(u_ 0\) is a Nash solution for \(J_ 0\); moreover the value of \(J_ 0\) in \(u_ 0=\lim_{h}u_ h\) is the limit of the values \(J_ h(u_ h)\), i.e. \(\lim_{h}J_ h(u_ h)=J_ 0(u_ 0)\). Stability properties of the \({\mathcal N}\)-convergence and its relations with \(\Gamma\)-limits and Yoshida approximations are also studied. A few examples are given: some of these are classical in Nash equilibrium problems, others are of the kind of ''optimal control problems''.