One central property of the S-matrix is its invariance under field redefinitions. I will discuss how the geometry of field space makes this invariance manifest. This geometric formulation also has practical consequences. The scattering amplitudes satisfy a geometric soft theorem.  Also, scattering amplitudes and the renormalization group equations for a theory of scalars and gauge bosons only depend on geometric quantities. I will apply these geometric techniques to calculate parts of the renormalization group equations for the Standard Model Effective Field Theory including operators up to mass dimension eight.