Given a cubic hypersurface X⊂P4, we study the existence of
Pfaffian representations of X, namely of 6×6 skew-symmetric matrices
of linear forms M such that X is defined by the equation Pf(M)=0. It was
known that such a matrix always exists whenever X is smooth. Here we prove
that the same holds whenever X is singular, hence that every cubic threefold
is Pfaffian