  Instantons, emerged in particle physics, have been intensely studied since
the 1970's and had an enormous impact in mathematics since then. In this paper,
we focus on one particular way in which mathematical physics has guided the
development of algebraic geometry in the past 40+ years. To be precise, we
examine how the notion of mathematical instanton bundles in algebraic geometry
has evolved from a class of vector bundles over the complex projective 3-space
both to a class of torsion free sheaves on projective varieties of arbitrary
dimension, and to a class of objects in the derived category of Fano
threefolds. The original results contained in this survey focus precisely on
the latter direction; in particular, we prove that the classical rank 2
instanton bundles over the projective 3-space are indeed instanton objects for
any suitable chamber in the space of Bridgeland stability conditions.
