Employing short laser pulses with a duration below 100 fs for changing magnetic state of magnetically-ordered media has developed into a distinct branch of magnetism —femtomagnetism which aims at controlling magnetization at ultimately short timescales. Among plethora of femtomagnetic phenomena, there is a class related to impact of femtosecond pulses on magnetic anisotropy of materials and nanostructures which defines orientation of magnetization, magnetic resonance frequencies and spin waves propagation. We present a review of main experimental results obtained in this field. We consider basic mechanisms responsible for a laser-induced change of various anisotropy types: magnetocrystalline, magnetoelastic, interfacial, shape anisotropy, and discuss specifics of these processes in magnetic metals and dielectrics. We consider several examples and describe features of magnetic anisotropy changes resulting from ultrafast laser-induced heating, impact of laser-induced dynamic and quasistatic strains and resonant excitation of electronic states. We also discuss perspectives of employing various mechanisms of laser-induced magnetic anisotropy change for enabling processes prospective for developing devices. We consider precessional magnetization switching for opto-magnetic information recording, generation of high-frequency strongly localized magnetic excitations and fields for magnetic nanotomography and hybrid magnonics, as well as controlling spin waves propagation for optically-reconfigurable magnonics. We further discuss opportunities which open up in studies of ultrafast magnetic anisotropy changes because of using short laser pulses in infrared and terahertz ranges.