Connes' bicentralizer problem is one of the most famous open problems in
the theory of type III factors. This conjecture was solved in the
amenable case by Haagerup in 1984, hence completing Connes' work on the
classification of amenable factors. But the general non-amenable case is
still open to this day, 40 years after Haagerup's celebrated work. In
this talk, I will first give an overview of Connes' bicentralizer
problem, its history and motivations, then I will present some recent
progress on this conjecture.