  We study scaling properties and topological aspects of the 2--d O(3)
non--linear $\sigma$--model on the lattice with the parametrized fixed point
action recently proposed by P.~Hasenfratz and F.~Niedermayer. The behavior of
the mass gap confirms the good properties of scaling of the fixed point action.
Concerning the topology, lattice classical solutions are proved to be very
stable under local minimization of the action; this outcome ensures the
reliability of the cooling method for the computation of the topological
susceptibility, which indeed reproduces the results of the field theoretical
approach. Disagreement is instead observed with a different approach in which
the fixed point topological charge operator is used: we argue that the
discrepancy is related to the ultraviolet dominated nature of the model.
