  In this chapter we review the rich behavior of polymer chains embedded in a
quenched random environment. We first consider the problem of a Gaussian chain
free to move in a random potential with short-ranged correlations. We derive
the equilibrium conformation of the chain using a replica variational ansatz,
and highlight the crucial role of the system's volume. A mapping is established
to that of a quantum particle in a random potential, and the phenomenon of
localization is explained in terms of the dominance of localized tail states of
the Schr\"odinger equation. We also give a physical interpretation of the
1-step replica-symmetry-breaking solution, and elucidate the connection with
the statistics of localized tail states. We proceeded to discuss the more
realistic case of a chain embedded in a sea of hard obstacles. Here, we show
that the chain size exhibits a rich scaling behavior, which depends critically
on the volume of the system. In particular, we show that a medium of hard
obstacles can be approximated as a Gaussian random potential only for small
system sizes. For larger sizes a completely different scaling behavior emerges.
Finally we consider the case of a polymer with self-avoiding (excluded volume)
interactions. In this case it is found that when disorder is present, the
polymer attains a conformation consisting of blobs connected by straight
segments. Using Flory type free energy arguments we analyze the statistics of
these conformational shapes, and show the existence of a
localization-delocalization transition as a function of the strength of the
self-avoiding interaction.
