  We extend the facial weak order from finite Coxeter groups to central
hyperplane arrangements. The facial weak order extends the poset of regions of
a hyperplane arrangement to all its faces. We provide four non-trivially
equivalent definitions of the facial weak order of a central arrangement: (1)
by exploiting the fact that the faces are intervals in the poset of regions,
(2) by describing its cover relations, (3) using covectors of the corresponding
oriented matroid, and (4) using certain sets of normal vectors closely related
to the geometry of the corresponding zonotope. Using these equivalent
descriptions, we show that when the poset of regions is a lattice, the facial
weak order is a lattice. In the case of simplicial arrangements, we further
show that this lattice is semidistributive and give a description of its
join-irreducible elements. Finally, we determine the homotopy type of all
intervals in the facial weak order.
