Flavour physics observables are excellent probes of new physics up to very high energy scales. Here we present FlavBit, the dedicated flavour physics module of the global-fitting package GAMBIT. FlavBit includes custom implementations of various likelihood routines for a wide range of flavour observables, including detailed uncertainties and correlations associated with LHCb measurements of rare, leptonic and semileptonic decays of B and D mesons, kaons and pions. It provides a generalised interface to external theory codes such as Superlso, allowing users to calculate flavour observables in and beyond the Standard Model, and then test them in detail against all relevant experimental data. We describe FlavBit and its constituent physics in some detail, then give examples from supersymmetry and effective field theory illustrating how it can be used both as a standalone library for flavour physics, and within GAMBIT.