Casimir Energies due to Matter Fields in $T^2$ and $T^2\mathrm{/}{Z}_2$ Compactifications

We calculate the Casimir energies due to matter fields with various boundary conditions along two compact directions in $T^2$ compactification. We discuss whether the Casimir energies generate attractive or repulsive forces. On the theories with extra dimensions, the Casimir energy plays a crucial role in the mechanism for stabilizing the size of extra dimensions. Finally we argue a procedure of the application to $Z_2$ orbifold.