Atomic interferometry was born recently, towards the end of the 1980s. Its development has been extremely fast, new techniques being pioneered independently and almost simultaneously in different laboratories all over the world. Nowadays, these techniques have reached a high level of sophistication, opening a wide area of fundamental and practical applications. In this paper the general architecture of interferometers in which matter waves are coherently manipulated is described. Various realizations of atom and molecule interferometers are reported, together with the major results obtained with each type of interferometer. Finally, new trends and perspectives are given. Whilst the techniques seem to be almost completely achieved, new developments are coming up, such as the use of new and non-ordinary sources. Forthcoming applications are numerous. They deal with the most fundamental aspects of quantum mechanics, with the metrology of fundamental constants, with the use of interferometers as very sensitive probes of external interactions and inertial effects, with atomic nanolithography, etc.