  We construct an $SO(10)$ grand unified theory in the formulation of
non-com-\break mutative geometry. The geometry of space-time is that of a
product of a continuos four dimensional manifold times a discrete set of
points. The properties of the fermionic sector fix almost uniquely the Higgs
structure. The simplest model corresponds to the case where the discrete set
consists of three points and the Higgs fields are ${\u {16}_s}\times
\overline{\u {16}}_s$ and ${\u {16}_s}\times {\u {16}_s} $. The requirement
that the scalar potential for all the Higgs fields not vanish imposes strong
restrictions on the vacuum expectation values of the Higgs fields and thus the
fermion masses. We show that it is possible to remove these constraints by
extending the number of discrete points and adding a singlet fermion and a ${\u
{16}_s}$ Higgs field. Both models are studied in detail.
