An Abelian differential induces a flat metric with saddle points such that the underlying Riemann surface can be realized as a polygon with edges pairwise identified by translation. Varying the shape of such polygons induces an SL(2,R) action on moduli spaces of Abelian differentials, called Teichmueller dynamics. Generic flat surfaces in an SL(2,R) orbit closure exhibit similar properties from the viewpoint of counting geodesics of bounded lengths, whose asymptotic growth rates satisfy a formula of Siegel-Veech type. In this talk I will give an introduction to this topic, with a focus on computing certain Siegel-Veech constants via intersection theory on moduli spaces.