The solvability of differential equations

It was a great surprise when Hans Lewy in 1957 presented a non-vanishing complex vector field that is not locally solvable. Actually, the vector field is the tangential Cauchy–Riemann operator on the boundary of a strictly pseudoconvex domain. Hörmander proved in 1960 that almost all linear partial differential equations are not locally solvable, because the necessary bracket condition is non-generic. This also has consequences for the spectral instability of non-selfadjoint semiclassical operators and the solvability of the Cauchy problem for non-linear analytic vector fields.
Nirenberg and Treves formulated their famous conjecture in 1970: that condition ($\Psi$) is necessary and sufficient for the local solvability of differential equations of principal type. Principal type essentially means simple characteristics, and condition ($\Psi$) only involves the sign changes of the imaginary part of the highest order terms along the bicharacteristics of the real part.
The Nirenberg–Treves conjecture was finally proved in 2003. We shall present the background, the main results, and some generalizations to non-principal type equations and systems of differential equations.