  The topological Tverberg conjecture was considered a central unsolved problem
of topological combinatorics. The conjecture asserts that for any integers
$r,d>1$ and any continuous map $f:\Delta\to\mathbb R^d$ of the
$(d+1)(r-1)$-dimensional simplex there are pairwise disjoint faces
$\sigma_1,\ldots,\sigma_r\subset\Delta$ such that $f(\sigma_1)\cap \ldots \cap
f(\sigma_r)\ne\emptyset$.
  The conjecture was proved for a prime power $r$. Recently counterexamples for
other $r$ were found. Analogously, the $r$-fold van Kampen-Flores conjecture
holds for a prime power $r$ but does not hold for other $r$. The arguments form
a beautiful and fruitful interplay between combinatorics, algebra and topology.
We present a simplified exposition accessible to non-specialists in the area.
We also mention some recent developments and open problems.
