We develop the theory of resolvent degree, introduced by Brauer cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilberts 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilberts 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.
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