  The Brownian Web (BW) is a family of coalescing Brownian motions starting
from every point in space and time $\R\times\R$. It was first introduced by
Arratia, and later analyzed in detail by T\'{o}th and Werner. More recently,
Fontes, Isopi, Newman and Ravishankar gave a characterization of the BW, and
general convergence criteria allowing either crossing or noncrossing paths,
which they verified for coalescing simple random walks. Later Ferrari, Fontes,
and Wu verified these criteria for a two dimensional Poisson Tree. In both
cases, the paths are noncrossing. In this thesis, we formulate new convergence
criteria for crossing paths, and verify them for non-simple coalescing random
walks (both discrete and continuous time) satisfying a finite fifth moment
condition. This is the first time convergence to the BW has been proved for
models with crossing paths. Several corollaries are presented, including an
analysis of the scaling limit of voter model interfaces that extends a result
of Cox and Durrett.
