  In the first part of this paper, we develop the theory of anisotropic
curvature measures for convex bodies in the Euclidean space. It is proved that
any convex body whose boundary anisotropic curvature measure equals a linear
combination of other lower order anisotropic curvature measures with
nonnegative coefficients is a scaled Wulff shape. This generalizes the
classical results by Schneider [Comment. Math. Helv. \textbf{54} (1979),
42--60] and by Kohlmann [Arch. Math. (Basel) \textbf{70} (1998), 250--256] to
the anisotropic setting. The main ingredients in the proof are the generalized
anisotropic Minkowski formulas and an inequality of Heintze--Karcher type for
convex bodies.
  In the second part, we consider the volume preserving flow of smooth closed
convex hypersurfaces in the Euclidean space with speed given by a positive
power $\alpha $ of the $k$th anisotropic mean curvature plus a global term
chosen to preserve the enclosed volume of the evolving hypersurfaces. We prove
that if the initial hypersurface is strictly convex, then the solution of the
flow exists for all time and converges to the Wulff shape in the Hausdorff
sense. The characterization theorem for Wulff shapes via the anisotropic
curvature measures will be used crucially in the proof of the convergence
result. Moreover, in the cases $k=1$, $n$ or $\alpha\geq k$, we can further
improve the Hausdorff convergence to the smooth and exponential convergence.
