The study of chaotic dynamical systems has highlighted their extreme sensitivity to initial conditions effectively ruling out any long term prediction of response. It is therefore quite surprising that, despite this sensitivity, under certain conditions two chaotic systems coupled together in some way, may synchronise their behaviour producing identical responses. This thesis mainly considers when synchronisation is accomplished with two identical dynamical systems coupled unidirectionally, i.e. with no feedback. In this way we aim to consider under what conditions a system may synchronise with an identical copy. As the coupling is altered the switch to convergence to the synchronised state can be viewed as a form of bifurcation. Initially, a trajectory will evolve within the higher dimensional coupled system phase space before eventually converging onto the invariant subset which is the synchronised state. In this thesis some of the main features of this property are discussed, and evidence will be given of the peculiarities that makes this bifurcation unique with respect to all other typical changes that occur in a dynamical system. This importance is assured by the evidence that when the dynamics in the invariant subset is chaotic new previously unseen phenomena occur, for instance the presence of very complex basins of attraction and peculiar intermittent behaviour taking place in the neighbourhood of the transition point. Moreover, extensive numerical work has been carried out to explain the underlying processes responsible of the behaviour of coupled systems possessing an invariant subset. Particular attention has been devoted to two apparently uncorrelated problems; first, the numerical occurrence of synchronised behaviour when the chaotic motion in the invariant subset is nevertheless unstable and second, the scaling of the distribution of the transient time before convergence to the invariant subset, when this is stable.