The operator eqution AX B = C (1) has been studied by several authors, but under the extra condition that the operators A and B have closed ranges. In this thesis, we present different results regarding the existence of solution and also the existence of positive solution to AXB = C without this extra hypothesis. The main goal of this thesis is to study the operator equation (1) where A , B and C are bounded linear operators defined on convenient Hilbert spaces. This kind of equation has been studied by several authors because of its multiple applications in different areas as, for example, control theory and sampling. The reader is referred to [6, 11, 18] and the references therein. However, in these works it is only considered the case in which A , B and C are matrices or have closed range. Our goal is to study equation (1) with arbitrary operators A , B and C . This consideration implies that some C , if and only if AA ? CB ? B = C for every inner inverses, A ? and B ? , of A and B , respectively. Recall that A ? (non-necessarily bounded) is an inner inverse of A if AA ? A = A . However, it is easy to see that this result fails if A , B have not closed range. In fact, for every operator A it holds AA ? AA ? A = A , but AXA = A is solvable if and only if A has closed range [14].