It is said that a long journey must begin with the first step. So it is with the systematic solution of Einstein field equations, the equations that Einstein developed in 1916 to describe the nature of space and time.
Unlike the equations that describe classical electrodynamics and even quantum theory, the fundamental equations of general relativity are nonlinear, a feature that has made them almost impossible to solve, except in approximate form. Of course, a number of solutions were discovered during the first fifty years of general relativity. In general, these were found by making extremely limiting assumptions concerning the form of the metric tensor gij, and then seeing whether there existed any solutions with the stipulated form. Most often there would exist no solution, but occasionally one was lucky. In this way a number of important solutions were discovered, not the least of which was the Schwarzschild solution that describes the gravitational field and spacetime structure associated with a non-rotating spherically symmetric body.
During the 1960's the emphasis switched from assumptions about the structure of the metric tensor gij, to assumptions about the structure of the Weyl conform tensor Cijkl, which is the part of the Riemann curvature tensor Rijkl, that does not vanish when the Einstein vacuum field equations are satisfied. The most noteworthy development of this period was the discovery in 1963 of the rotating Schwarzschild solution of Roy Kerr.
I became aware of the Kerr solution in 1966, and I spent nine months trying to convince myself that it was indeed a solution of the Einstein equations. The Nobel prize winner S. Chandrasekhar told me that he also spent nine months doing the same. It was just a very difficult thing to do! The effort convinced me that I had to find some way to make it obvious that Kerr's metric satisfied the Einstein field equations. Therefore, I began a study which culminated in the discovery of what I called the complex potential formulation of the Einstein field equations, which I published early in 1968. This sequel to my book Time Travel is a description of my complex potential approach, a famous conjecture of Robert Geroch in 1971, the subsequent study of the infinitesimal Geroch group by Kinnersley and Chitre, and the proof of the Geroch conjecture by Isidore Hauser and Frederick J. Ernst.
The approach that Hauser and I employed to prove the Geroch conjecture for stationary axisymmetric spacetimes, where elliptic differential equations were involved, was eventually generalized by us to cover the more difficult case of colliding gravitational plane waves, where hyperbolic differential equations are involved.