Using newly developed cohomology theories for algebraic varieties, it has become possible to construct refinements of classical numerical invariants, such as Euler characteristics, degrees of Chern classes and intersection numbers, to have values in the Grothendieck-Witt ring of quadratic forms. We will give an overview of the foundations in motivic homotopy theory that make this possible and illustrate with some concrete examples, such as counting lines or other rational curve on algebraic varieties.