In this paper, we describe in detail the BMV (Bir\'efringence Magn\'etique du
Vide) experiment, a novel apparatus to study the propagation of light in a
transverse magnetic field. It is based on a very high finesse Fabry-Perot
cavity and on pulsed magnets specially designed for this purpose. We justify
our technical choices and we present the current status and perspectives.Comment: To be published in the European Physical Journal