  We study the structure and representation theory of affine wreath product
algebras and their cyclotomic quotients. These algebras, which appear naturally
in Heisenberg categorification, simultaneously unify and generalize many
important algebras appearing in the literature. In particular, special cases
include degenerate affine Hecke algebras, affine Sergeev algebras (degenerate
affine Hecke-Clifford algebras), and wreath Hecke algebras. In some cases,
specializing the results of the current paper recovers known results, but with
unified and simplified proofs. In other cases, we obtain new results, including
proofs of two open conjectures of Kleshchev and Muth.
