The capacity of reaction network system to exhibit two or more steady states has been the focus of considerable recent work. The question of multistationarity is closely related to that of injectivity of the corresponding vector field. In this talk we give an overview of some old and new results on injectivity and multistationarity in vector fields associated with interaction networks, under more or less general assumptions on the nature of the network and the kinetic laws. This is joint work with Murad Banaji.