The space of kinematic invariants for the scattering of n massless particles 'knows’ very little about the physical properties of scattering matrices. In this talk, the space of Riemann spheres with n marked points is shown to be more naturally suited for building tree-level scattering matrices. The two spaces are connected by the ‘scattering equations’ and compact formulas for the complete tree-level S-matrices of scalars, gluons and gravitons can be derived.