Let \(k\) be a commutative ring, \(H\) a \(k\)-bialgebra, \(A\) a right \(H\)-comodule algebra and denote by \({\mathcal M}^H_A\) (\(_A{\mathcal M}^H\)) the category of relative right (left) Hopf modules. Let \(\tau\) be an element of the convolution algebra \(\text{Hom} (H,\text{End}(A))\). Then the authors construct the algebra \(A^\tau\) by twisting the multiplication on \(A\), and the right \(A^\tau\)-module \(M^\tau\), for \(M\in {\mathcal M}^H_A\). They give necessary and sufficient conditions for \(A^\tau\) to be an \(H\) comodule algebra and for \(F_\tau:{\mathcal M}^H_A\to {\mathcal M}^H_{A^\tau}\), \(M\mapsto M^\tau\), \(f\mapsto f\) to be a functor. \(A^\tau\) is called a twisted algebra of \(A\), and it is proved that if \(\tau\) is convolution invertible, then \(F_\tau\) is an isomorphism. A left hand version of the twisting is also defined, and similarly there is an isomorphism between the categories of left Hopf modules, provided that \(H\) is a Hopf algebra with a bijective antipode. One of the main results states that if \(\sigma\) is an invertible cocycle, then the crossed product \(A\#_\sigma H\) proves to be an invertible twisting of \(B\otimes H\), this leading to a new description of cleft extensions. It is also shown that if \(k\) is a field, \(\tau\) is invertible and \(^{*\text{rat}}\neq 0\), then \(A\# H^{*\text{rat}}\simeq A^\tau\# H^{*\text{rat}}\) as algebras. As a corollary to these results, a duality theorem for crossed products is obtained.