Many of the problems encountered in several branches of mathematics, engineering, and physics require the existence and uniqueness of solutions of operator equations of the form Tx = b, where b is a given vector in some Hilbert space X and T is a densely defined linear operator on X. It is obvious that the above equation has a unique solution if and only if the equation Tx = 0 has only the trivial solution x = 0. It is also well-known that if T has a closed range, then there exists a solution of Tx = b if and only if ⟨y, b⟩ = 0, where ⟨·, ·⟩ denotes the inner product on X and y is any solution of the adjoint equation T ∗ y = 0. This project will focus on the perturbation of closed range operators T on a Hilbert space X with emphasis on the stability of the closedness of the range under relatively bounded perturbations. The effect of these perturbations on the nullity and the deficiency of T in addition to the solutions of the operator equation above will also be considered. The theory of the minimum modulus of T and that of its generalized inverse are expected to play a significant role in this study.