  A simplicial graph is said to be (coarsely) Helly if any collection of
pairwise intersecting balls has non-empty (coarse) intersection. (Coarsely)
Helly groups are groups acting geometrically on (coarsely) Helly graphs. Our
main result is that finitely generated groups that are hyperbolic relative to
(coarsely) Helly subgroups are themselves (coarsely) Helly. One important
consequence is that various classical groups, including toral relatively
hyperbolic groups, are equipped with a CAT(0)-like structure -- they act
geometrically on spaces with convex geodesic bicombing. As a means of proving
the main theorems we establish a result of independent interest concerning
relatively hyperbolic groups: a `relatively hyperbolic' description of
geodesics in a graph on which a relatively hyperbolic group acts geometrically.
In the other direction, we show that for relatively hyperbolic (coarsely) Helly
groups their parabolic subgroups are (coarsely) Helly as well. More generally,
we show that `quasiconvex' subgroups of (coarsely) Helly groups are themselves
(coarsely) Helly.
