We derive a new method for initial-state collinear showering in Monte-Carlo event generators which is based on the use of unintegrated parton correlation functions. Combined with a previously derived method for final-state showering, the method solves the problem of treating both the hard scattering and the evolution kernels to be used in arbitrarily non-leading order. Although we only treat collinear showering, so that further extensions are needed for QCD, we have discovered several new results: (1) It is better to generate exact parton kinematics in the hard scattering rather than with the subsequent parton showering, and similarly at each step of the showering. (2) Parton showering is then done conditionally on the exact 4-momentum of the initiating parton. (3) We obtain a factorization for structure functions in terms of parton correlation functions so that parton kinematics can be treated exactly from the beginning. (4) We obtain two factorization properties for parton correlation functions, one in terms of ordinary parton densities and one, suitable for event generation, in terms of parton correlation functions themselves.