Lattice quantum field theories containing fennions can be formulated in a chirally invariant way provided long-range interactions are introduced. I establish that in weak-coupling perturbation theory such a lattice theory is renormalizable when the corresponding continuum theory is, and that the continuum theory is indeed recovered in the perturbative continuum limit. In the strong-coupling limit of these theories one is led to study an effective Hamiltonian describing a Heisenberg antiferromagnet with long-range interactions. Using blockspin renormalization group methods I find a critical rate of falloff of the interactions, approximately as inverse distance squared, which separates a nearest-neighbor-antiferromagnetic phase from a phase displaying identifiable long-range effects. I point out a duality-type symmetry which is present in some block-spin calculations.