Interferometry as a space-time process is described, together with its topology. Starting from this viewpoint, a convenient unified formalism for the phase shifts which arise in particle interferometry is developed. This formalism is based on a covariant form of Hamilton`s action principle and Lagrange`s equations of motion. It will be shown that this Lorentz invariant formalism yields a simple perturbation theoretic expression for the general phase shift that arises in matter-wave interferometry. The Lagrangian formalism is compared with the more usual formalism based on the wave propagation vector and frequency. The resulting formalism will be used to analyse the Sagnac effect, gravitational field measurements, and several Aharonov-Bohm-like topological phase shifts. Several topological interferometric experiments using particles of non-zero rest mass are discussed. These experiments involve the use of electrons, neutrons and neutral atoms. Neutron experiments will be emphasised. 45 refs., 15 figs.