If $C$ is a smooth curve over an algebraically closed field $k$ of characteristic $p$, then the structure of the maximal prime to $p$ quotient of the ��tale fundamental group is known by analytic methods. In this paper, we discuss the properties of the fundamental group that can be deduced by purely algebraic techniques. We describe a general reduction from an arbitrary curve to the projective line minus three points, and show what can be proven unconditionally about the maximal pro-nilpotent and pro-solvable quotients of the prime-to-$p$ fundamental group. Included is an appendix which treats the tame fundamental group from a stack-theoretic perspective.

26 pages. Significant revision; errors corrected, various points clarified