  The purpose of this paper is to develop a theory of bimonads and Hopf monads
on arbitrary categories thus providing the possibility to transfer the
essentials of the theory of Hopf algebras in vector spaces to more general
settings. There are several extensions of this theory to {\em monoidal}
categories which in a certain sense follow the classical trace. Here we do not
pose any conditions on our base category but we do refer to the monoidal
structure of the category of endofunctors on any category $\A$ and by this we
retain some of the combinatorial complexity which makes the theory so
interesting. As a basic tool we use distributive laws between monads and
comonads (entwinings) on $\A$: we define a {\em bimonad} on $\A$ as an
endofunctor $B$ which is a monad and a comonad with an entwining $\lambda:BB\to
BB$ satisfying certain conditions. This $\lambda$ is also employed to define
the category $\A^B_B$ of (mixed) $B$-bimodules. In the classical situation, an
entwining $\lambda$ is derived from the twist map for vector spaces. Here this
need not be the case but there may exist special distributive laws $\tau:BB\to
BB$ satisfying the Yang-Baxter equation ({\em local prebraidings}) which induce
an entwining $\lambda$ and lead to an extension of the theory of {\em braided
Hopf algebras}. An antipode is defined as a natural transformation $S:B\to B$
with special properties and for categories $\A$ with limits or colimits and
bimonads $B$ preserving them, the existence of an antipode is equivalent to $B$
inducing an equivalence between $\A$ and the category $\A^B_B$ of
$B$-bimodules. This is a general form of the {\em Fundamental Theorem} of Hopf
algebras.
