This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. r 1 (z − y j ) = 0. Note that z = y j is possible but cannot hold for all j since µ > 0.Define Y j ⊂ X j for j = 1, . . . , r via y j ∈ relint conv Y j . We claim that r 1 aff Y j = ∅. Otherwise there is a point v ∈ r 1 aff Y j . Let ·, · denote the standard scalar product, so x, x = x 2 , for instance. Then z − v, z − y j > 0 if y i = z (because y j is the closest point to z in conv Y j ) Imre Bárány