It has been known for some time that the six Painlevé equations are reductions of the self-dual Yang-Mills equations under the action of various subgroups of the conformal group. The twistor theory of this result is reviewed, and also its application to the construction of classical solutions and special geometries. Two generalizations are described, which are related by an extended form of Harnad's duality. One gives a twistor description of the solution of the general isomonodromy problem with any number of irregular singularities; the second corresponds to a problem with two singularities, a regular one at the origin and an irregular one at infinity. The two are related by a simple operation on the corresponding bundle over twistor space.