  We describe how quartic ($\lambda \phi^4$) inflation with non-minimal
coupling to gravity is realized in realistic supersymmetric $SO(10)$ models. In
a well-motivated example the $16-\overline{16}$ Higgs multiplets, which break
$SO(10)$ to $SU(5)$ and yield masses for the right-handed neutrinos, provide
the inflaton field $\phi$. Thus, leptogenesis is a natural outcome in this
class of $SO(10)$ models. Moreover, the adjoint (45-plet) Higgs also acquires a
GUT scale value during inflation so that the monopole problem is evaded. The
scalar spectral index $n_s$ is in good agreement with the observations and $r$,
the tensor to scalar ratio, is predicted for realistic values of GUT parameters
to be of order $10^{-3}-10^{-2}$.
