  Arratia, and later T\'oth and Werner, constructed random processes that
formally correspond to coalescing one-dimensional Brownian motions starting
from every space-time point. We extend their work by constructing and
characterizing what we call the {\em Brownian Web} as a random variable taking
values in an appropriate (metric) space whose points are (compact) sets of
paths. This leads to general convergence criteria and, in particular, to
convergence in distribution of coalescing random walks in the scaling limit to
the Brownian Web.
