The Fornax cluster provides a uniquely compact laboratory in which to study the detailed history of early-type galaxies and the role played by the environment in driving their evolution and their transformation from late-type galaxies. Using the superb capabilities of the Multi Unit Spectroscopic Explorer on the Very Large Telescope, high-quality integral-field spectroscopic data were obtained for the inner regions of all the bright (mB ≤ 15) galaxies within the virial radius of Fornax. The stellar haloes of early-type galaxies are also covered out to about four effective radii. State-of-the-art stellar dynamical and population modelling allows characterising the disc components of fast-rotating early-type galaxies, constraining radial variations in the stellar initial-mass functions and measuring the stellar age, metallicity, and α-element abundance of stellar haloes in cluster galaxies. This paper describes the sample selection, observations, and overall goals of the survey, and provides initial results based on the spectroscopic data, including the detailed characterisation of stellar kinematics and populations to large radii; decomposition of galaxy components directly via their orbital structure; the ability to identify globular clusters and planetary nebulae, and derivation of high-quality emission-line diagnostics in the presence of complex ionised gas.