  Let \Delta be a finite building (or, more generally, a thick spherical and
locally finite building). The chamber graph G(\Delta), whose edges are the
pairs of adjacent chambers in \Delta, is known to be q-regular for a certain
number q=q(\Delta). Our main result is that G(\Delta) is q-connected in the
sense of graph theory.
  Similar results are proved for the chamber graphs of Coxeter complexes and
for order complexes of geometric lattices.
