  We survey the basic theory of non-commutative K3 surfaces, with a particular
emphasis to the ones arising from cubic fourfolds. We focus on the problem of
constructing Bridgeland stability conditions on these categories and we then
investigate the geometry of the corresponding moduli spaces of stable objects.
We discuss a number of consequences related to cubic fourfolds including new
proofs of the Torelli theorem and of the integral Hodge conjecture, the
extension of a result of Addington and Thomas and various applications to
hyperk\"ahler manifolds.
  These notes originated from the lecture series by the first author at the
school on "Birational Geometry of Hypersurfaces", Palazzo Feltrinelli -
Gargnano del Garda (Italy), March 19-23, 2018.
