The X-J model is a strong-coupling limit of both the natural tight binding model of perovskite superconductors and the Anderson lattice model of heavy fermions. The application of the model to perovskite superconductivity is straightforward, but its use to describe heavy fermions is more speculative. The straight-line motion of charge carriers in the model is sympathetic to antiferromagnetic correlations along the path traversed, although the motion destroys the long-range antiferromagnetic order by exchanging the two sublattices in passing. Antiferromagnetism is destroyed in both the square lattice geometry relevant to a CuO2 plane and the triangular geometry relevant to an isolated layer of CeAl3. A paramagnetic phase with shorter range correlations than suggested by the Heisenberg model seems preferred by the charge-carrier motion in these two-dimensional examples.