Ferroelectric tunnel junctions combine the phenomena of quantum-mechanical tunnelling and switchable spontaneous polarisation of a nanometre-thick ferroelectric film into novel device functionality. Switching the ferroelectric barrier polarisation direction produces a sizable change in resistance of the junction - a phenomenon known as the tunnelling electroresistance effect. From a fundamental perspective, ferroelectric tunnel junctions and their version with ferromagnetic electrodes, i.e., multiferroic tunnel junctions, are testbeds for studying the underlying mechanisms of tunnelling electroresistance as well as the interplay between electric and magnetic degrees of freedom and their effect on transport. From a practical perspective, ferroelectric tunnel junctions hold promise for disruptive device applications. In a very short time, they have traversed the path from basic model predictions to prototypes for novel non-volatile ferroelectric random access memories with non-destructive readout. This remarkable progress is to a large extent driven by a productive cycle of predictive modelling and innovative experimental effort. In this review article, we outline the development of the ferroelectric tunnel junction concept and the role of theoretical modelling in guiding experimental work. We discuss a wide range of physical phenomena that control the functional properties of ferroelectric tunnel junctions and summarise the state-of-the-art achievements in the field.