In this thesis we discuss progress towards proving homological mirror symmetry for the genus 2 curve in an abelian variety. We describe a fully faithful functor from the bounded derived category of coherent sheaves on the genus 2 curve to the Fukaya-Seidel (FS) category of an SYZ mirror constructed via methods of Auroux-Abouzaid-Katzarkov for hypersurfaces in toric varieties. In particular, part of the FS category construction involves counting curves.