  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 \'etale 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.
