Raynaud's theory of theta divisors links the algebraic fundamental group of a smooth projective curve over an algebraically closed field of characteristic p to the geometry of a certain canonically constructed theta divisor on the jacobian. For example, building on the work of Raynaud and Pop-Saidi, Tamagawa recently showed that there are at most a finite number of isomorphism classes of hyperbolic curves over a finite field with a given fundamental group. In this talk, after introducing the theory of theta divisors and fundamental groups of curves in characteristic p, I'll present the results of Jilong Tong (a recent student of Raynaud) on refined geometric properties of the theta divisor (especially in characteristics 2 and 3) and the corresponding implications for the fundamental group.