  In this paper the authors introduce a new notion called the quantum wreath
product, which is the algebra $B \wr_Q \mathcal{H}(d)$ produced from a given
algebra $B$, a positive integer $d$, and a choice $Q=(R,S,\rho,\sigma)$ of
parameters. Important examples {that arise from our construction} include many
variants of the Hecke algebras, such as the Ariki-Koike algebras, the affine
Hecke algebras and their degenerate version, Wan-Wang's wreath Hecke algebras,
Rosso-Savage's (affine) Frobenius Hecke algebras, Kleshchev-Muth's affine
zigzag algebras, and the Hu algebra that quantizes the wreath product $\Sigma_m
\wr \Sigma_2$ between symmetric groups.
  In the first part of the paper, the authors develop a structure theory for
the quantum wreath products. Necessary and sufficient conditions for these
algebras to afford a basis of suitable size are obtained. Furthermore, a
Schur-Weyl duality is established via a splitting lemma and mild assumptions on
the base algebra $B$. Our uniform approach encompasses many known results which
were proved in a case by case manner. The second part of the paper involves the
problem of constructing natural subalgebras of Hecke algebras that arise from
wreath products. Moreover, a bar-invariant basis of the Hu algebra via an
explicit formula for its extra generator is also described.
