This talk outlines three distinct aspects of higher Lie characters. The k-th root enumerator of a finite group G is an integer valued function on G, which counts the number of k-th roots of each element. A long-standing open problem is to classify the finite groups for which the k-th root enumerator is a proper (non-virtual) character for all k. Extending results of Scharf and Thibon, we show that for all classical Weyl groups, all k-th root enumerators are proper. The proof is constructive, and presents the root enumerator as a multiplicity-free sum of higher Lie characters of classical types. The decomposition of higher Lie characters into irreducibles is an eighty years old problem of Thrall. Coarse versions of this problem were studied by Desarmenien-Wachs, Reiner-Webb, and others. We present an asymptotic solution. Finally, we apply higher Lie characters to prove a permutation-statistics refinement of an elegant result comparing the numbers of permutations with cycles of only even and only odd lengths. Based on joint works with Pal Hegedus, Yuval Roichman and Natalia Tsilevich.