The concepts of volume averaging are briefly reviewed and a fundamental weakness of the standard averaging approach is examined. A generalized volume averaging method (use of a space-time varying averaging volume) is presented which reduces to the standard approach under specific limiting conditions. The generalized approach is valid whenever there exists suitably smooth averaging volumes, Theorems relating the averages of derivative to derivatives of averages are presented and proved. The macroscopic equations of an arbitrary extensive property are developed for statistically, locally homogeneous, multiphase media, and a probabilistic interpretation of the results is presented.