While there have been quantum leaps recently in our ability to calculate
scattering amplitudes with massless legs in four dimensions in recent
years, not nearly so much has been done in higher dimensions. Apart from
theoretical interest, higher dimensional amplitudes feature promptly in
four dimensional loop computations and are directly relevant for string
theory. A major obstacle is that recent developments depend heavily on
four dimensional spinor helicity methods. I will present a complete
extension of these methods to all higher dimensions. Simple applications
yield series of vanishing amplitudes in Yang-Mills and gravity theories
and quite generic on-shell supersymmetry Ward identities.