THIS IS A JOINT ALGEBRAIC TOPOLOGY / TOPOLOGY SEMINAR.  PLEASE NOTE DIFFERENT TIME AND LOCATION.   We survey some new methods and results on existence and on topological classification of  actions of connected, compact Lie groups on manifolds. This will include construction of some counterexamples to conjectures about transformation groups of aspherical manifolds and investigation of generalized multiaxial actions of unitary groups. (Here generalized multiaxial topological actions are those for which the isotropy subgroups are sub-unitary groups, as occurs for certain linear representations.) The topological results on abundance of multiaxial actions contrast sharply with classical smooth results. This talk reports on joint works with Shmuel Weinberger and Min Yan.