In this lecture, I will discuss three different problems from Discrete Geometry, the Topological Tverberg Problem, the Colored Tverberg Problem, and the Grünbaum Hyperplane Problem. These problems have many things in common: They are easy to state, and may look harmless, They have very nice and classical configuration spaces, they may be attacked by ``Equivariant Obstruction Theory'', this solves the problems --- but only partially, which leads us to ask more questions, look for new tools ... and this yields surprising new results. (Joint work with Pavle Blagojevic, Florian Frick, Albert Haase, and Benjamin Matschke)