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)