Kontsevich and Manin have introduced the moduli spaces of stable maps
as background spaces on which to count curves with fixed properties
in projective manifolds. In this talk I will describe a new construction
of the Kontsevich-Manin spaces for rational curves, starting from the
simpler example of the Grassmannian of lines in the projective space. This
new construction allows us to understand the structure of their
cohomology rings in a natural, geometric way. Time permitting, I will explain
how this construction can be formalized by setting a family of
different stability conditions on maps from curves to the projective space.
This talk is based on joint work with Andrei Mustata.