  In this paper, we present upper bounds for the depth of some classes of
polyhedra, including: polyhedra with finite fundamental group, polyhedra $P$
with abelian or free $\pi_1(P)$ and finitely generated
$H_i(tilde{P};\mathbb{Z}$, 2-dimensional polyhedra with abelian or free
fundamental group, and 2-dimensional polyhedra with elementary amenable
fundamental group $G$ with finite cohomological dimension $cd(G)$. Furthermore,
we provide some examples to show that some of these bounds are sharp.
