Neutron-induced fission reactions play a crucial role in a variety of fields of fundamental and applied nuclear science. In basic nuclear physics they provide important information on properties of nuclear matter, while in nuclear technology they are at the basis of present and future reactor designs. Finally, there is a renewed interest in fission reactions in nuclear astrophysics due to the multi-messenger observation of neutron star mergers and the important role played by fission recycling in r-process nucleosynthesis. Although studied for several decades, many fundamental questions still remain on fission reactions, while modern applications and the development of more reliable nuclear models require high-accuracy and consistent experimental data on fission cross sections and other fission observables. To address these needs, an extensive fission research programme has been carried out at the n_TOF neutron time-of-flight facility at CERN during the last 18 years, taking advantage of the high energy resolution, high luminosity and wide energy range of the neutron beam, as well as of the detection and data acquisition systems designed for this purpose. While long-lived isotopes are studied on the 185 m long flight-path, the recent construction of a second experimental area at a distance of about 19 m has opened the way to challenging measurements of short-lived actinides. This article provides an overview of the n_TOF experimental programme on neutron-induced fission reactions along with the main characteristics of the facility, the various detection systems and data analysis techniques used. The most important results on several major and minor actinides obtained so far and the future perspectives of fission measurements at n_TOF are presented and discussed.