A user's guide to the topological Tverberg conjecture
The well-known topological Tverberg conjecture was considered a central unsolved problem of topological combinatorics. The conjecture asserts that for each integers $r,d>1$ and each continuous map $f\colon\Delta\to\mathbb R^d$ of the $(d+1)(r-1)$-dimensional simplex $\Delta$ there are pairwise disjoint subsimplices $\sigma_1,\ldots,\sigma_r\subset\Delta$ such that $f(\sigma_1)\cap \ldots \cap f(\sigma_r)\ne\emptyset$. 
A proof for a prime power $r$ was given by I. Barany, S. Shlosman, A. Szucs, M. Ozaydin and A. Volovikov in 1981-1996. A counterexample for other $r$ was found in a series of papers by M. Ozaydin, M. Gromov, P. Blagojevic, F. Frick, G. Ziegler, I. Mabillard and U. Wagner, most of them recent. The arguments form a beautiful and fruitful interplay between combinatorics, algebra and topology. In this expository paper we present a simplified explanation of easier parts of the arguments, accessible to non-specialists in the area, and give reference to more complicated parts.