The talk deals with the classification of smooth projective curves, abelian varieties and Severi-Brauer varieties over a field k up to isomorphism in the motivic homotopy category H(k). For this we need the motivic homotopy invariance of the genus of a curve and the motivic homotopical behaviour of A^1-rigid schemes. Furthermore Nikita A. Karpenko classified Severi-Brauer varieties up to motivic equivalence as Chow motives. To use this, we have to make a connection between motivic homotopy theory and Chow motives.