Angle-resolved photoemission spectroscopy allows direct visualization and experimental determination of the electronic structure of crystals in the momentum space, including the precise characterization of the Fermi surface and the superconducting order parameter. It is thus particularly suited for investigating multiband systems such as the Fe-based superconductors. In this review, we cover several aspects of these recently discovered materials that have been addressed by this technique, with a special emphasis on their superconducting gap and their Fermi surface topology. We provide sufficient experimental evidence to support the reliability and the consistency of the angle-resolved photoemission spectroscopy measurements over a wide range of material compositions. ; Comment: 42 pages, 17 figures. Accepted for publication in Reports on Progress in Physics. This version is the authors' one. After publication, the final version with high resolution figures will be available at http://iopscience.iop.org/0034-4885/