  In this paper we make an overview of results relating the recent
"discoveries" in differential geometry, such as higher structures and
differential graded manifolds with some natural problems coming from mechanics.
We explain that a lot of classical differential geometric constructions in the
context can be conveniently described using the language of Q-structures, and
thus Q-structure preserving integrators are potentially of great use in
mechanics. We give some hints how the latter can be constructed, and formulate
some open problems.
  Since the text is intended both to mathematics and mechanics communities, we
tried to make it accessible to non-geometers as well.
