General linear methods were originally introduced in the hope that they
would provide a unifying basis for the study of a wide variety of existing
methods. A second hope was that this formulation would point the way to
new practical methods. This lecture will survey early work by the author
and by Ernst Hairer and Gerhard Wanner in which the theories of order and
stability for this broad class of methods methods were established.
Alongside these general and theoretical developments, the search for
methods suitable for practical implementation has shown only limited
success. The author now believes that there is some advantage in
restricting attention to methods possessing ``RK stability" (that is,
methods which behave like Runge-Kutta methods, from the linear stability
point of view). The derivation of methods in this restricted class, using
techniques discovered in collaboration with Will Wright, will be
discussed.