  We introduce a general method to induce Bridgeland stability conditions on
semiorthogonal components of triangulated categories. In particular, we prove
the existence of Bridgeland stability conditions on the Kuznetsov component of
the derived category of Fano threefolds and of cubic fourfolds. As an
application, in the appendix, written jointly with Xiaolei Zhao, we give a
variant of the proof of the Torelli theorem for cubic fourfolds by Huybrechts
and Rennemo.
