The paper introduces the quasi-minimal residual (QMR) method, a novel approach for solving non-Hermitian linear systems. The QMR method is designed to overcome the limitations of the biconjugate gradient (BCG) method, which can exhibit numerical instabilities and breakdowns. The QMR method is based on a look-ahead version of the nonsymmetric Lanczos algorithm, which prevents breakdowns by relaxing the biorthogonality condition when a breakdown is imminent. The QMR iterates are defined by a quasi-minimal residual property, ensuring smooth convergence and stable recovery of BCG iterates. The paper also derives an error bound for the QMR method, similar to those for the generalized minimal residual (GMRES) algorithm, and discusses the incorporation of preconditioning techniques to enhance the method's performance. Numerical experiments are presented to validate the effectiveness of the QMR method.The paper introduces the quasi-minimal residual (QMR) method, a novel approach for solving non-Hermitian linear systems. The QMR method is designed to overcome the limitations of the biconjugate gradient (BCG) method, which can exhibit numerical instabilities and breakdowns. The QMR method is based on a look-ahead version of the nonsymmetric Lanczos algorithm, which prevents breakdowns by relaxing the biorthogonality condition when a breakdown is imminent. The QMR iterates are defined by a quasi-minimal residual property, ensuring smooth convergence and stable recovery of BCG iterates. The paper also derives an error bound for the QMR method, similar to those for the generalized minimal residual (GMRES) algorithm, and discusses the incorporation of preconditioning techniques to enhance the method's performance. Numerical experiments are presented to validate the effectiveness of the QMR method.