In this talk we will discuss the Cauchy-Riemann equations on domains in complex manifolds with positive or negative curvature. We will also report some recent new results on the $L^2$ closed range property for $\bar{\partial}$ on an annulus between two pseudoconvex domains, when the inner domain is not smooth. In particular, we show the Hausdor property of the $L^2$ Dolbeault cohomology group on a domain between a ball and a bi-disc, the so-called Chinese Coin problem. We also give characterizaation of Lipschitz domains with holes through their Dolbeault cohomology groups. (joint work with Debraj Chakrabarti, Siqi Fu, and Christine Laurent-Thiebaut).