Summary: We establish existence and uniqueness results for the two-dimensional dissipative quasi-geostrophic (QG) equation with initial data in a Besov space \(B^r_{p,q}\) or in the function space \(B^{r, \delta}_{p,q}\) introduced in this paper. Attention is focused on the QG equation with a critical or supercritical fractional power of the Laplacian for which the dissipation is insufficient to balance the nonlinearity. The function space \(B^{r, \delta}_{p,q}\) is a natural generalization of the Besov space \(B^r_{p,q}\) and it allows study of the two-dimensional QG equation in a functional setting that is close to a borderline Besov space.