  This article investigates structural, geometrical, and topological
characterizations and properties of weakly modular graphs and of cell complexes
derived from them. The unifying themes of our investigation are various
`nonpositive curvature' and `local-to-global' properties and characterizations
of weakly modular graphs and their subclasses. Weakly modular graphs have been
introduced as a far-reaching common generalization of median graphs (and more
generally, of modular and orientable modular graphs), Helly graphs, bridged
graphs, and dual polar graphs occurring under different disguises in several
seemingly-unrelated fields of mathematics: Metric graph theory, Geometric group
theory, Incidence geometries and buildings, Theoretical computer science and
combinatorial optimization. We give a local-to-global characterization of
weakly modular graphs and their subclasses in terms of simple connectedness of
associated triangle-square complexes and specific local combinatorial
conditions. In particular, we revisit characterizations of dual polar graphs by
Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are
precisely the clique-Helly graphs with simply connected clique complexes. With
$l_1$-embeddable weakly modular and sweakly modular graphs we associate
high-dimensional cell complexes, having several strong topological and
geometrical properties (contractibility and the CAT(0) property). Their cells
have a specific structure: they are basis polyhedra of even
$\triangle$-matroids in the first case and orthoscheme complexes of gated dual
polar subgraphs in the second case. We resolve some open problems concerning
subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture
about CAT(0) metric on the orthoscheme complexes of modular lattices; we answer
Chastand's question about prime graphs for pre-median graphs.
