The study of topological centres has been a very active field in Banach algebra theory and abstract harmonic analysis for many years. I shall report on recent progress concerning the solution of various topological centre problems, on the one hand, and the use of such results as a tool, on the other hand. Particular emphasis will be placed on the following: the positive solution, for a large class of compact non-metrizable groups, of the CecchiniZappa conjecture [1981] on the centre of the bidual of the Fourier algebra (joint work with M. Filali and M. Sangani Monfared) note that, as shown by V. Losert, the conjecture fails for the compact metrizable group SU(3); the positive solution of the GhahramaniLau conjecture [1994] on the topological centres of the bidual of the measure algebra over a locally compact group (joint work with Stefano Ferri, V. Losert, J. Pachl and J. Steprans); the negative solution of a question raised by Laulger [1996] on the structure of certain multipliers on von Neumann algebras (joint work with Z. Hu and Z.-J. Ruan); the negative solution of FarhadiGhahramani’s multiplier problem [2007]; topological centres for group actions and their relation to the number of invariant means for the action (joint work with J. Pachl and J. Steprans); topological centres and invariant means for algebras over locally compact quantum groups (joint work with Z. Hu and Z.-J. Ruan).