It is unknown whether free-by-cyclic groups can always be distinguished from one another by their finite quotients, i.e. whether they are always profinitely rigid within the class of free-by-cyclic groups. After surveying the context of this question, I will describe joint work with Pawel Piwek in which we show that all free-by-cyclic groups with centre are profinitely rigid in this sense. I shall also explain why groups of this form resolve the old question of whether one can simultaneously minimise the rank (number of generators) and deficiency of a finitely presented group. A key idea in the proofs is that instead of approaching these groups as mapping tori, one should view them as the fundamental groups of a certain type of non-positively 2-complex.