A general, non-self-adjoint, complex, elliptic boundary-value problem is approximated by a system of complex, finite-difference equations and solved by SOR techniques. Since transformations of the complex difference equations to real equations lead to systems which lack diagonal dominance, the difference equations are solved iteratively as a complex system. A brief account of complex SOR theory is provided. This includes block as well as point methods and, for the complex point Jacobi matrix, the distribution of the eigenvalues and the proof that there is one eigenvalue pair with maximum modulus are given. Important considerations in the application of complex methods—accuracy, factors influencing convergence and the automatic determination of convergence factors—are examined.