  We develop the twisting construction for locally compact quantum groups. A
new feature, in contrast to the previous work of M. Enock and the second
author, is a non-trivial deformation of the Haar measure. Then we construct
Rieffel's deformation of locally compact quantum groups and show that it is
dual to the twisting. This allows to give new interesting concrete examples of
locally compact quantum groups, in particular, deformations of the classical
$az+b$ group and of the Woronowicz' quantum $az+b$ group.
