  We study the directed polymer model for general graphs (beyond $\mathbb Z^d$)
and random walks. We provide sufficient conditions for the existence or
non-existence of a weak disorder phase, of an $L^2$ region, and of very strong
disorder, in terms of properties of the graph and of the random walk. We study
in some detail (biased) random walk on various trees including the Galton
Watson trees, and provide a range of other examples that illustrate
counter-examples to intuitive extensions of the $\mathbb Z^d$/SRW result.
