We will use geometric Langlands theory to solve two problems concerning number fields. One is Serre's question of whether there exist motives over $\Bbb Q$ with motivic Galois group of type $E_8$ or $G_2$; the other is the concrete question of whether there are Galois extensions of $\Bbb Q$ with Galois group $E_8(p)$ or $G_2(p)$ (the finite simple groups of Lie type), for sufficiently large primes $p$. The answer to both questions is ``YES". Please note the change of day for this week's colloquium.