In characteristic zero, Castelnuovo proved that a unirational surface is rational. In positive characteristic, this fails. We discuss the plethora of non-rational, often general-type, surfaces that are unirational in positive characteristic. In 1977, Shioda conjectured that these surfaces are classified by a cohomological property: supersingularity. I will demonstrate a counterexample to this conjecture. We will need a new obstruction that uses ideas from the study of hyperbolicity.