  In this paper we investigate the problem of a long self-avoiding polymer
chain immersed in a random medium. We find that in the limit of a very long
chain and when the self-avoiding interaction is weak, the conformation of the
chain consists of many ``blobs'' with connecting segments. The blobs are
sections of the molecule curled up in regions of low potential in the case of a
Gaussian distributed random potential or in regions of relatively low density
of obstacles in the case of randomly distributed hard obstacles. We find that
as the strength of the self-avoiding interaction is increased the chain
undergoes a delocalization transition in the sense that the appropriate free
energy per monomer is no longer negative. The chain is then no longer bound to
a particular location in the medium but can easily wander around under the
influence of a small perturbation. For a localized chain we estimate
quantitatively the expected number of monomers in the ``blobs'' and in the
connecting segments.
