  The classical equations of motion of the perfect lattice action in
asymptotically free $d=2$ spin and $d=4$ gauge models possess scale invariant
instanton solutions. This property allows the definition of a topological
charge on the lattice which is perfect in the sense that no topological defects
exist. The basic construction is illustrated in the $d=2$ O(3) non--linear
$\sigma$--model and the topological susceptibility is measured to high
precision in the range of correlation lengths $\xi \in (2 - 60)$. Our results
strongly suggest that the topological susceptibility is not a physical quantity
in this model.
