  This is the first in a series of papers devoted to generalisations of
statistical loop models. We define a lattice model of $U_q(\mathfrak{sl}_n)$
webs on the honeycomb lattice, for $n \ge 2$. It is a statistical model of
closed, cubic graphs with certain non-local Boltzmann weights that can be
computed from spider relations. For $n=2$, the model has no branchings and
reduces to the well-known O($N$) loop model introduced by Nienhuis. In the
general case, we show that the web model possesses a particular point, at
$q=e^{i \pi/(n+1)}$, where the partition function is proportional to that of a
$\mathbb{Z}_n$-symmetric chiral spin model on the dual lattice. Moreover, under
this equivalence, the graphs given by the configurations of the web model are
in bijection with the domain walls of the spin model. For $n=2$, this
equivalence reduces to the well-known relation between the Ising and O($1$)
models. We define as well an open $U_q(\mathfrak{sl}_n)$ web model on a simply
connected domain with a boundary, and discuss in particular the role of defects
on the boundary.
