It is well known that linear and nonlinear stability concepts are equivalent for linear multistep methods in their one-leg formulation. This result is extended to Runge-Kutta methods. In particular, it is shown here that given an irreducible rational function R(z) whose degrees of numerator and denominator are at most s which has order of approximation \(p\geq 1\) to exp(z) and has \(| R(z)| \leq 1\) for all Re(z)\(\leq 0\) then there exists an s-stage algebraically stable Runge- Kutta method of order p with R(z) as the stability function.