THIS IS A SPECIAL ALGEBRAIC TOPOLOGY SEMINAR.  PLEASE NOTE DIFFERENT DATE, TIME AND LOCATION.   In this talk we present an evolution of equivariant topology methods in Combinatorial Geometry. We start with:  (a) the Topological Radon's theorem, an application of the Borsuk-Ulam theorem, and proceed, via non-planarity of K_{3,3}, to (b) the Topological Tverberg and the Weak Colored Tverberg theorem for primes, which are applications of Dold's theorem, to continue with (c) the Topological Tverberg for prime powers, an application beyond Dold's theorem based on the connectivity and localization theorem for elementary abelian groups, to finally ask: "What needs to be done in the case of Barany-Larman conjecture and Nandakumar & Ramana-Rao problem when all the previously known methods fail?"