  Under some general assumptions, we construct the scaling limit of open
clusters and their associated counting measures in a class of two dimensional
percolation models. Our results apply, in particular, to critical Bernoulli
site percolation on the triangular lattice and to the critical FK-Ising model
on the square lattice. Fundamental properties of the scaling limit, such as
conformal covariance, are explored. As an application to Bernoulli percolation,
we obtain the scaling limit of the largest cluster in a bounded domain. We also
apply our results to the critical, two-dimensional Ising model, obtaining the
existence and uniqueness of the scaling limit of the magnetization field, as
well as a geometric representation for the continuum magnetization field which
can be seen as a continuum analog of the FK representation.
