This paper presents a new class of fractional order Runge–Kutta (FORK) methods for numerically approximating the solution of fractional differential equations (FDEs). We construct explicit and implicit FORK methods for FDEs by using the Caputo generalized Taylor series formula. Due to the dependence of fractional derivatives on a fixed base point, in the proposed method, we had to modify the right-hand side of the given equation in all steps of the FORK methods. Some coefficients for explicit and implicit FORK schemes are presented. The convergence analysis of the proposed method is also discussed. Numerical experiments are presented to clarify the effectiveness and robustness of the method.