  The talk presented at ICMP 97 focused on the scaling limits of critical
percolation models, and some other systems whose salient features can be
described by collections of random lines. In the scaling limit we keep track of
features seen on the macroscopic scale, in situations where the short--distance
scale at which the system's basic variables are defined is taken to zero. Among
the challenging questions are the construction of the limit, and the
explanation of some of the emergent properties, in particular the behavior
under conformal maps as discussed in [LPS 94]. A descriptive account of the
project, and some related open problems, is found in ref. [A] and in [AB]
(joint work with A. Burchard) where tools are developed for establishing a
curve--regularity condition which plays a key role in the construction of the
limit. The formulation of the scaling limit as a random Web measure permits to
formulate the question of uniqueness of measure(s) describing systems of random
curves satisfying the conditions of independence, Euclidean invariance, and
regularity. The uniqueness question remains open; progress on it could shed
light on the purported universality of critical behavior and the apparent
conformal invariance of the critical measures. The random Web yields also
another perspective on some of the equations of conformal field theory which
have appeared in this context, such as the equation proposed by J. Cardy [C].
