  We consider equivariant versions of the motivic Chern and Hirzebruch
characteristic classes of a quasi-projective toric variety, and extend many
known results from non-equivariant to the equivariant setting. The
corresponding generalized equivariant Hirzebruch genus of a torus-invariant
Cartier divisor is also calculated. Further global formulae for equivariant
Hirzebruch classes are obtained in the simplicial context by using the Cox
construction and the equivariant Lefschetz-Riemann-Roch theorem. Alternative
proofs of all these results are given via localization at the torus fixed
points in equivariant $K$- and homology theories. In localized equivariant
$K$-theory, we prove a weighted version of a classical formula of Brion for a
full-dimensional lattice polytope. We also generalize to the context of motivic
Chern classes the Molien formula of Brion-Vergne. Similarly, we compute the
localized Hirzebruch class, extending results of Brylinski-Zhang for the
localized Todd class.
  We also elaborate on the relation between the equivariant toric geometry via
the equivariant Hirzebruch-Riemann-Roch and Euler-Maclaurin type formulae for
full-dimensional simple lattice polytopes. Our results provide generalizations
to arbitrary coherent sheaf coefficients, and algebraic geometric proofs of
(weighted versions of) the Euler-Maclaurin formulae of Cappell-Shaneson,
Brion-Vergne, Guillemin, etc., via the equivariant Hirzebruch-Riemann-Roch
formalism. Our approach, based on motivic characteristic classes, allows us to
obtain such Euler-Maclaurin formulae also for (the interior of) a face, or for
the polytope with several facets removed. We also prove such results in the
weighted context, and for Minkovski summands of the given full-dimensional
lattice polytope. Some of these results are extended to local Euler-Maclaurin
formulas for the tangent cones at the vertices of the given lattice polytope.
