  Relatively few families of Bell inequalities have previously been identified.
Some examples are the trivial, CHSH, I_{mm22}, and CGLMP inequalities. This
paper presents a large number of new families of tight Bell inequalities for
the case of many observables. For example, 44,368,793 inequivalent tight Bell
inequalities other than CHSH are obtained for the case of 2 parties each with
10 2-valued observables. This is accomplished by first establishing a
relationship between the Bell inequalities and the facets of the cut polytope,
a well studied object in polyhedral combinatorics. We then prove a theorem
allowing us to derive new facets of cut polytopes from facets of smaller
polytopes by a process derived from Fourier-Motzkin elimination, which we call
triangular elimination. These new facets in turn give new tight Bell
inequalities. We give additional results for projections, liftings, and the
complexity of membership testing for the associated Bell polytope.
