  The procedure for obtaining integrable vertex models via reflection matrices
on the square lattice with open boundaries is reviewed and explicitly carried
out for a number of two- and three-state vertex models. These models include
the six-vertex model, the 15-vertex $A_2^{(1)}$ model and the 19-vertex models
of Izergin-Korepin and Zamolodchikov-Fateev. In each case the eigenspectra is
determined by application of either the algebraic or the analytic Bethe ansatz
with inhomeogeneities. With suitable choices of reflection matrices, these
vertex models can be associated with integrable loop models on the same
lattice. In general, the required choices {\em do not} coincide with those
which lead to quantum group-invariant spin chains. The exact solution of the
integrable loop models -- including an $O(n)$ model on the square lattice with
open boundaries -- is of relevance to the surface critical behaviour of
two-dimensional polymers.
