  Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf
algebras that contain Hecke algebras $H_{\bf q}(W)$ as (left) coideal
subalgebras. Our Hecke-Hopf algebras ${\bf H}(W)$ have a number of
applications. In particular they provide new solutions of quantum Yang-Baxter
equation and lead to a construction of a new family of endo-functors of the
category of $H_{\bf q}(W)$-modules. Hecke-Hopf algebras for the symmetric group
are related to Fomin-Kirillov algebras, for an arbitrary Coxeter group $W$ the
"Demazure" part of ${\bf H}(W)$ is being acted upon by generalized braided
derivatives which generate the corresponding (generalized) Nichols algebra.
