In this talk we introduce a new numerical method for computing the Hamiltonian Schur form of a 2n-by-2n Hamiltonian matrix M that has no purely imaginary eigenvalues. We demonstrate the properties  of the new method by showing its performance for the benchmark collection of continuous-time algebraic Riccati equations. Despite the fact that no complete error analysis for the method is yet available, the numerical results indicate that if no eigenvalues of M are close to the imaginary axis then the method computes the exact Hamiltonian Schur form of a nearby Hamiltonian matrix and thus is numerically strongly backward stable. The new method  is of complexity O(n^3) and hence it solves a long-standing open problem in numerical analysis.