$n$-Categories in Algebraic Topology

After a brief sketch of Batanin's definition of weak $n$-categories,
we describe conjectures and theorems relating various special classes
of $n$-categories to algebraic topology.  For example: weak $n$-groupoids
are believed to be a model for homotopy $n$-types; strict $n$-groupoids are
known to be a model for homotopy $n$-types with trivial Postnikov invariants 
(but possibly nontrivial action of $\pi_1$ on the higher homotopy groups);
stable strict $n$-groupoids are the same as $n$-term chain complexes of
abelian groups.  More interestingly, certain $n$-categories that are not 
$n$-groupoids play a role in higher-dimensional knot theory.