This monograph contains a study of the global Cauchy problem for the Yang-Mills equations on (6 1) and higher dimensional Minkowski space, when the initial data sets are small in the critical gauge covariant Sobolev space H (n 4)/2A. Regularity is obtained through a certain ""microlocal geometric renormalization"" of the equations which is implemented via a family of approximate null Croenstrom gauge transformations. The argument is then reduced to controlling some degenerate elliptic equations in high index and non-isotropic Lp spaces, and also proving some bilinear estimates in specially constructed square-function spaces.