The development of new analytic and computational methods in the perturbative approach to Yang-Mills theories is of great importance to the understanding of LHC physics. Twistor string theory, a form of string theory defined on Penrose's twistor space, may provide a very efficient and powerful alternative to the usual Feynman diagram expansion of field theory. I survey the recent construction of a family of new twistor string theories which are free from world-sheet anomalies and give the space-time spectra of Einstein supergravities, with second order field equations instead of the higher derivative conformal supergravities that arose from earlier twistor strings.
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