As an excuse to talk about one of my favorite results, I thought I'd put this out there (even though I've already mentioned this to Tyler privately).

Abhyankar conjectured that the the collection of finite quotients of the étale fundamental group of the affine line in characteristic $p$ are exactly the quasi-$p$-groups. This was proved by Raynaud (as mentoned above). A slightly more complicated statement (for general curves) was quickly thereafter proved by Harbater.

Here's an even more interesting (to my mind) result:

Suppose $X$ a geometrically connected, projective variety of dimension over any field $K$ of positive characteristic. Suppose $L$ an ample line bundle on $X$, $D$ a closed subscheme of dimension less than $n$, and $S$ a $0$-dimensional subscheme of the regular locus of $X$ not meeting $D$. Then there exists a positive integer $r$ and an $(n+1)$-tuple of linearly independent sections of $L^{\otimes r}$ with no common zero such that the induced finite morphism $f : X \to P^n_K$ of $K$-schemes meets the following conditions.

(1) If $H$ denotes the hyperplane at infinity, then $f$ is étale away from $H$.

(2) The image $f(D)$ is contained in $H$.

(3) The image $f(S)$ does not meet $H$.

This was proved by Abhyankar in dimension $1$, and the general result is due to Kedlaya. The proof is just gorgeous; it's even simpler than his first paper on the subject, which only works for infinite fields $K$.

This says something pretty remarkable: even though, in characteristic $0$, affine spaces are simply connected, in positive characteristic, *every* variety contains a Zariski open that is an étale cover of affine space! (Katz uses this kind of trick in his notes on Weil II.)