  We survey recent developments in the study of torus equivariant motivic Chern
and Hirzebruch characteristic classes of projective toric varieties, with
applications to calculating equivariant Hirzebruch genera of torus-invariant
Cartier divisors in terms of torus characters, as well as to general
Euler-Maclaurin type formulae for full-dimensional simple lattice polytopes. We
present recent results by the authors, emphasizing the main ideas and some key
examples. This includes global formulae for equivariant Hirzebruch classes in
the simplicial context proved by localization at the torus fixed points, a
weighted versions of a classical formula of Brion, as well as of the Molien
formula of Brion-Vergne.
  Our Euler-Maclaurin type formulae provide generalizations to arbitrary
coherent sheaf coefficients 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, e.g., to obtain such Euler-Maclaurin
formulae also for (the interior of) a face. We obtain such results also in the
weighted context, and for Minkovski summands of the given full-dimensional
lattice polytope.
