Abstract: Veech groups arise as stabilizers of Teichmüller disks associated to translation surfaces, and lattice Veech groups give Teichmüller curves in moduli space. In this talk, I will introduce the basic definitions and examples of Veech groups, focusing on their interpretation through flat geometry, affine automorphisms, and Teichmüller theory. I will then discuss the genus two case, where McMullen’s classification relates primitive Teichmüller curves to real multiplication and Hilbert modular surfaces. Finally, I will explain how genus-two Veech groups act on first homology through the symplectic representation, and how this provides a useful linear-algebraic viewpoint on their structure.