This talk will be an elementary survey of results on differentials, connections, and differential equations in the algebro-geometric context. Motivated by some simple examples, we will first discuss known results on when these differential equations have solutions complex analytically. After some discussion of the peculiarities of differentials in characteristic p, we will state the p-curvature conjecture, a famous unresolved conjecture of Grothendieck's which says when these complex analytic solutions are actually algebraic.