  We set up a strategy for studying large families of logarithmic conformal
field theories by using the enlarged symmetries and non--semi-simple
associative algebras appearing in their lattice regularizations (as discussed
in a companion paper). Here we work out in detail two examples of theories
derived as the continuum limit of XXZ spin-1/2 chains, which are related to
spin chains with supersymmetry algebras gl($n|n$) and gl($n+1|n$),
respectively, with open (or free) boundary conditions in all cases. These
theories can also be viewed as vertex models, or as loop models. Their
continuum limits are boundary conformal field theories (CFTs) with central
charge $c=-2$ and $c=0$ respectively, and in the loop interpretation they
describe dense polymers and the boundaries of critical percolation clusters,
respectively. We also discuss the case of dilute (critical) polymers as another
boundary CFT with $c=0$. Within the supersymmetric formulations, these boundary
CFTs describe the fixed points of certain nonlinear sigma models that have a
supercoset space as the target manifold, and of Landau-Ginzburg field theories.
The submodule structures of indecomposable representations of the Virasoro
algebra appearing in the boundary CFT, representing local fields, are derived
from the lattice. A central result is the derivation of the fusion rules for
these fields.
