“Blackness theory” is described as a class of procedures for matching a high order transport approximation in one region to a low order approximation in a second region. The matching conditions are presented as a generalization of the Marshak boundary conditions. The blackness coefficients necessary in setting up the conditions are defined and tables are given for slab geometry. A method which allows all regions to be treated by means of the blackness coefficients is developed and applied to two region cells. Numerical results are compared with other approximations in situations typical of those encountered in resonance capture and thermal utilization calculations.