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For many years,symplectic Runge-Kutta methods have been the mainstay of numerical methods for Hamiltonian problems.One of the secrets of the success is the canonical requirement biaij+bjaji=bibj, which guarantees that quadratic invariants are conserved.For general linear methods there is a related condition with similar consequences. This talk will examine the relationship between so-called G-symplectic integrators and symplectic Runge-Kutta methods.It will be shown how to construct G-symplectic methods of order at least four and to construct starting and finishing methods required for practical implementation. Numerical experiments suggest that the new methods are able to achieve accuracy comparable to that of Runge-Kutta methods.A tentative expla nation of this surprisingly good performance will be offered.