  We study the analogue of tautological rings of fibre bundles in the context
of fibrations with Poincar\'e fibre, i.e. the ring obtained by fibre
integrating powers of the fibrewise Euler class. We approach this ring of
characteristic classes using rational homotopy theory. More specifically, we
discuss a cdga model for the universal fibration and chain level
representatives of fibre integration and the fibrewise Euler class. This is
particularly useful for rationally elliptic spaces and as an example we compute
the tautological ring for complex projective spaces and products of odd
spheres.
