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A review of the Contractor Renormalization (CORE) method, as a systematicderivation of the low energy effective hamiltonian, is given, with emphasis onits differences and advantages over traditional perturbative (weak/stronglinks) real space RG. For the low energy physics of the 2D Hubbard model, wederive the plaquette bosons (projected SO(5)) model which connects themicroscopic model to phases and phenomenology of high-Tc cuprates. For theS=1/2 Pyrochlore and Kagome antiferromagnets, the effective hamiltonianspredict spin-disordered, lattice symmetry breaking, ground states with a largedensity of low energy singlets as found by exact diagonalization of smallclusters.Comment: For the proceedings of the conference on 'Effective Models for  Low-Dimensional Strongly Correlated Systems', Peyresq, September 2005.  Peyresq Conference. 11 pages, 9 Figure

Computing Effective Hamiltonians of Doped and Frustrated Antiferromagnets By Contractor Renormalization