A number of results are given on the relationship between equivariant and non-equivariant bundles and their ification. The bundles dealt with are the projections to orbits $E\to E/\Pi $, where $\Pi $ is a normal subgroup of a compact Lie group $\Gamma $ and $E$ is a $\Pi $-free $\Gamma $-space. The base space has an action by $G=\Gamma /\Pi $, and such bundles are ified by a $G$-space $B(\Pi ,\Gamma )$. Information about the homotopy type of this $G$-space gives information about the set of equivalent es of such bundles with base a given $G$-space $X$. The bundle theory considerably simplifies when $G$ acts freely on $X$, and the main theme is the study of the transformation on bundles theories induced by athe natural projection $EG\times X\to X$.