  The Brownian Web (BW) is the random network formally consisting of the paths
of coalescing one-dimensional Brownian motions starting from every space-time
point in ${\mathbb R}\times{\mathbb R}$. We extend the earlier work of Arratia
and of T\'oth and Werner by providing characterization and convergence results
for the BW distribution, including convergence of the system of all coalescing
random walkssktop/brownian web/finale/arXiv submits/bweb.tex to the BW under
diffusive space-time scaling. We also provide characterization and convergence
results for the Double Brownian Web, which combines the BW with its dual
process of coalescing Brownian motions moving backwards in time, with forward
and backward paths ``reflecting'' off each other. For the BW, deterministic
space-time points are almost surely of ``type'' $(0,1)$ -- {\em zero} paths
into the point from the past and exactly {\em one} path out of the point to the
future; we determine the Hausdorff dimension for all types that actually occur:
dimension 2 for type $(0,1)$, 3/2 for $(1,1)$ and $(0,2)$, 1 for $(1,2)$, and 0
for $(2,1)$ and $(0,3)$.
