This thesis deals with the application of the astrometry technique, consisting in measuring the position of a star in the plane of the sky, for the discovery and characterisation of extra-solar planets. It is feasible only with a very high measurement precision, which motivates the use of space observatories, the development of new ground-based astronomical instrumentation and of innovative data analysis methods: The study of Sun-like stars with substellar companions using CORALIE radial velocities and HIPPARCOS astrometry leads to the determination of the frequency of close brown dwarf companions and to the discovery of a dividing line between massive planets and brown dwarf companions; An observation campaign employing optical imaging with a very large telescope demonstrates sufficient astrometric precision to detect planets around ultra-cool dwarf stars and the first results of the survey are presented; Finally, the design and initial astrometric performance of PRIMA, a new dual-feed near-infrared interferometric observing facility for relative astrometry is presented.