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In this talk we present a new colored version of Tverberg's theorem about partitions of a finite point set in R^d into rainbow groups whose convex hulls intersect. This settles the famous Barany-Larman conjecture (1992) for primes minus one, and asymptotically in general. We also give a transversal generalization of the new colored Tverberg theorem, which has unifying character, since it contains the center transversal theorem and the ham sandwich theorem as special cases. The proofs use equivariant topology and contain calculations of independent interest. This is joint work with Pavle V. Blagojević and Günter M. Ziegler.