Runge-Kutta methods are typically represented in the so-called Butcher form. The more general Shu-Osher form for explicit
Runge-Kutta methods was introduced later to facilitate analysis of stability properties like monotonicity, and positivity. We review the recent generalization of the Shu-Osher form, and show how it can be used to facilitate understanding and development of improved Runge-Kutta methods, in terms of stability, solvability, implementation, and memory efficiency. Furthermore, we show how the modified Shu-Osher form is a natural tool for studying other classes ofone-step methods, such as extrapolation and deferred correction methods, that are not typically thought of as Runge-Kutta methods.