The GMRES algorithm has been discussed to solve the linear system Ax = b for application in a flow solver. Besides the conventional GMRES algorithm, several approaches are discussed which supposedly reduce the computational costs of the method, and reduce the number of iterations needed to satisfy the convergence criteria. GMRES is one of the most popular methods for solving non-symmetric systems. The method is robust, i.e. the solution of the least squares problem always exists. GMRES does have full recurrences since the basis of the Krylov subspace needs to be stored completely. Also, orthogonalisation of new basis vectors becomes increasingly more expensive when the iterations proceed. GMRES minimizes the residual norm, as a result the residual norm decreases monotonically and the convergence is smooth.