In 1798, Gauss described a composition law on the space of SL_2(Z) equivalence classes of binary quadratic forms with fixed discriminant D. This law turned this set into a group, which turned out to be the narrow class group of the field Q(\sqrt{D}).  However, this composition law was complicated and seemed unmotivated.  200 years later, an interpretation was found, which led to the higher composition laws.  I will discuss these spaces, where they come from and what has been done with them.