Journal of Statistical Physics, Vol. 103, Nos. 5/6, 2001. p. 915-944.
In the last decade there has been an enormous progress in the mathematical understanding of one-dimensional polymer measures, which are path measures that suppress self-intersections. We are currently in the situation that many interesting questions have either been answered, or that essential new ideas are needed. In this survey paper, we discuss the most relevant results, open questions, and heuristics.