This monograph contains the worked out and expanded notes of the lecture
series I presented at the 37–th Probability Summer School in SaintFlour,
8–21 July 2007. The goal I had set myself for these lectures was to
provide an up-to-date account of some key developments in the mathematical
theory of polymer chains, focusing on a number of models that are at the
heart of the subject. In order to achieve this goal, I decided to limit myself
to single polymers living on a lattice, to consider only models for which a
transparent picture has emerged, and from the latter select those that lead to
challenging open questions capable of attracting future research. Needless to
say, my choice was influenced by my personal taste and involvement