A Veech surface is a translation surface whose (affine) automorphism group is as large as possible. While a generic point of a Veech surface equidistributes under the action of its automorphism group, there is an exceptional finite set of points with finite orbits. These periodic points appear throughout Teichmüller dynamics, such as in counting holomorphic sections of families of Riemann surfaces and in blocking problems on billiard tables. In this talk we first describe joint work with Zawad Chowdhury, Samuel Everett and Destine Lee that gives an algorithm that computes the set of periodic points for a given Veech surface. We then describe work that classifies the periodic points of Prym eigenforms, an infinite family of Veech surfaces in genera 2, 3 and 4.