  A graph is Helly if every family of pairwise intersecting combinatorial balls
has a nonempty intersection. We show that weak Garside groups of finite type
and FC-type Artin groups are Helly, that is, they act geometrically on Helly
graphs. In particular, such groups act geometrically on spaces with convex
geodesic bicombing, equipping them with a nonpositive-curvature-like structure.
That structure has many properties of a CAT(0) structure and, additionally, it
has a combinatorial flavor implying biautomaticity. As immediate consequences
we obtain new results for FC-type Artin groups (in particular braid groups and
spherical Artin groups) and weak Garside groups, including e.g.\ fundamental
groups of the complements of complexified finite simplicial arrangements of
hyperplanes, braid groups of well-generated complex reflection groups, and
one-relator groups with non-trivial center. Among the results are:
biautomaticity, existence of EZ and Tits boundaries, the Farrell-Jones
conjecture, the coarse Baum-Connes conjecture, and a description of higher
order homological and homotopical Dehn functions. As a mean of proving the
Helly property we introduce and use the notion of a (generalized) cell Helly
complex.
