  We classify the first few brane tilings on a genus 2 Riemann surface and
identify their toric Calabi-Yau moduli spaces. These brane tilings are
extensions of tilings on the 2-torus, which represent one of the largest known
classes of 4d N=1 superconformal field theories for D3-branes. The
classification consists of 16 distinct genus 2 brane tilings with up to 8
quiver fields and 4 superpotential terms. The Higgs mechanism is used to relate
the different theories.
