Instanton bundles were defined as the algebraic counterpart that permitted to solve a central problem in Yang-Mills theory. By means of a theorem by Hitchin, there only exist two projective varieties on which this link can be done meaningfully: the projective space P^3 and the flag threefold F of point-lines. Whereas the case of P^3 has been thoroughly studied and led to many breakthroughs in Algebraic Geometry (construction of moduli spaces, monads,...) much less work has been devoted to the flag threefold.

The aim of this talk is twofold: firstly, I will recall the main results obtained during the last 40 years for instanton bundles on the projective space P^3. Secondly, I will explain our recent contributions in this area in the case of the flag threefold F, underlying the similarities and differences with the former case. This is joint work with V. Antonelli, F. Malaspina and S. Marchesi.