Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by simulation on a computer. One hopes that, by performing simulations on larger and larger lattices, while making the lattice spacing smaller and smaller, one will be able to recover the behaviour of the continuum theory.
Just as in all lattice models, numerical simulation gives access to field configurations that are not accessible to perturbation theory, such as solitons. Likewise, non-trivial vacuum states can be discovered and probed.
The method is particularly appealing for the quantization of a gauge theory. Most quantization methods keep Poincaré invariance manifest but sacrifice manifest gauge symmetry by requiring gauge fixing. Only after renormalization can gauge invariance be recovered. Lattice field theory differs from these in that it keeps manifest gauge invariance, but sacrifices manifest Poincaré invariance— recovering it only after renormalization. The articles on lattice gauge theory and lattice QCD explore these issues in greater detail.