As announced in [12], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study convolability and invertibility of Lagrangian conic submanifolds of the symplectic groupoid T * G. We also identify those Lagrangian which correspond to equivariant families parametrized by the unit space G (0) of homogeneous canonical relations in (T * Gx \ 0) x (T * G x \ 0). This allows us to select a subclass of Lagrangian distributions on any Lie groupoid G that deserve the name of Fourier integral G-operators (G-FIO). By construction, the class of G-FIO contains the class of equivariant families of ordinary Fourier integral operators on the manifolds Gx, x $\in$ G (0). We then develop for G-FIO the first stages of the calculus in the spirit of Hormander's work. Finally, we work out an example proving the efficiency of the present approach for studying Fourier integral operators on singular manifolds.