Abstract: I will present a new notion of stability for periodic orbits in polygonal billiards. A periodic orbit of a polygonal billiard is $\lambda$-stable if there is a periodic orbit for the corresponding pinball billiard which converges to it as $\lambda\to1$. This notion of stability is unrelated to the notion introduced by Galperin, Stepin and Vorobets. We will derive sufficient and necessary conditions for a periodic orbit to be $\lambda$-stable and determine completely the $\lambda$-stable periodic orbits in integrable polygons. This is joint work with Serge Troubetzkoy.