Cauchy's Theorem for complex integrals with a condition

Question

Does Cauchy's Theorem for complex integrals hold if I have the following scenario:

I integrate a function $f$ which is analytic (holomorphic) over a region $A$ except for a point $z_0$.

1. If that point $z_0$ is inside the region, when does Cauchy's Theorem hold?
2. If that point $z_0$ is on the boundary of that region, when does Cauchy's Theorem hold?

Thanks.