In this paper we consider the problem of solving the Cauchy-Riemann equation with prescribed support. More precisely, let X be a complex manifold of complex dimension n and Ω ⊂ X a subdomain of X. We ask the following questions:Let T be a ∂-closed (r, 1)-current, 0 ≤ r ≤ n, on X with support contained in Ω, does there exist a (r, 0)-current on X, with support contained in Ω, such that ∂S = T ?If moreover T = f is a smooth form or a C k form or an L p loc form, can we find g with support contained in Ω and with the same regularity as f such that ∂g = f ?This leads us to introduce the Dolbeault cohomology groups with prescribed support in Ω. Let us denote by H r,1 Ω,∞ (X) the quotient spaceIn the same way, we define H r,1 Ω,C k (X), H r,1 Ω,L p loc (X) and H r,1 Ω,cur (X) for the C k , L p loc and the current category.The cohomology groups H r,1 Ω,∞ (X), H r,1 Ω,C k (X) ,H r,