This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if
π
:
H
→
H
¯
\pi :H \to \overline H
is a Hopf algebra epimorphism which is split as a coalgebra map, then
H
H
is algebra isomorphic to
A
#
σ
H
A{\# _\sigma }H
, a crossed product of
H
H
with the left Hopf kernel
A
A
of
π
\pi
. Given any crossed product
A
#
σ
H
A{\# _\sigma }H
with
H
H
(weakly) inner on
A
A
, then
A
#
σ
H
A{\# _\sigma }H
is isomorphic to a twisted product
A
τ
[
H
]
{A_\tau }[H]
with trivial action. Finally, if
H
H
is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of
A
A
implies that of
A
#
σ
H
A{\# _\sigma }H
; in particular this is true if the (weak) action of
H
H
is inner.