In this paper we study the properties of Drinfeld's twisting for
finite-dimensional Hopf algebras. We determine how the integral of the dual to
a unimodular Hopf algebra $H$ changes under twisting of $H$. We show that the
classes of cosemisimple unimodular, cosemisimple involutive, cosemisimple
quasitriangular finite-dimensional Hopf algebras are stable under twisting. We
also prove the cosemisimplicity of a coalgebra obtained by twisting of a
cosemisimple unimodular Hopf algebra by two different twists on two sides (such
twists are closely related to biGalois extensions), and describe the
representation theory of its dual. Next, we define the notion of a
non-degenerate twist for a Hopf algebra $H$, and set up a bijection between
such twists for $H$ and $H^*$. This bijection is based on Miyashita-Ulbrich
actions of Hopf algebras on simple algebras. It generalizes to the
non-commutative case the procedure of inverting a non-degenerate skew-symmetric
bilinear form on a vector space. Finally, we apply these results to
classification of twists in group algebras and of cosemisimple triangular
finite-dimensional Hopf algebras in positive characteristic, generalizing the
previously known classification in characteristic zero.