A sharp affine $L_p$ Sobolev inequality for functions on Euclidean $n$-space is established. This new inequality is significantly stronger than (and directly implies) the classical sharp $L_p$ Sobolev inequality of Aubin and Talenti, even though it uses only the vector space structure and standard Lebesgue measure on $\Bbb R^n$. For the new inequality, no inner product, norm, or conformal structure is needed; the inequality is invariant under all affine transformations of $\Bbb R^n$.