Resolvent degree is an invariant of a branched cover which quantifies how "hard" is it to specify a point in the cover given a point under it in the base. Historically, this was applied to the branched cover $\P^n/S_{n-1} \to \P^n/S_n$, from the moduli of degree n polynomials with a specified root to the moduli of degree n polynomials. Classical enumerative problems and congruence subgroups provide two additional sources of branched covers to which this invariant applies. In ongoing joint work with Benson Farb, and with Benson Farb and Mark Kisin, we develop the theory of resolvent degree as an extension of Buhler and Reichstein's theory of essential dimension. We apply this theory to systematize an array of classical results and to relate the complexity of seemingly different problems such as finding roots of polynomials, lines on cubic surfaces, and level structures on Jacobians.