The Cauchy-Riemann equations are fundamental in one and several complex variables. Holomorphic functions, for example, satisfy the homogeneous Cauchy-Riemann equations. In complex Euclidean space of dimension n≥2, the necessary and sufficient condition for the solvability of the Cauchy-Riemann equations is that the domain is pseudoconvex. In this talk we relate pseudoconvexity with the vanishing of L2 Dolbeault cohomology groups. On a pseudoconvex domain in a complex manifold, the L2 Dolbeault cohomology might not even be Hausdorff. Recent results on the L2 closed range property for ∂¯ on an annulus between two pseudoconvex domains will be discussed. One can even characterize such domains through the spectral theory of their L2 Dolbeault cohomology groups, thus hearing pseudoconvexity of the boundary. (Joint work with Debraj Chakrabarti, Siqi Fu, and Christine Laurent-Thiébaut).