Motivated by counting problems for polygonal billiards and more generally for linear flows on surfaces, Veech introduced what is now known as the Siegel-Veech transform on the moduli space of abelian differentials, in analogy with the Siegel transform arising from the space of unimodular lattices in R^n. It this talk, I will present the proof that the Siegel-Veech transform of a compactly supported continuous function is square-integrable with respect to the Masur-Veech measure and give applications to bounding error terms for counting problems for saddle connections. This is joint work with Jayadev Athreya and Howard Masur.