The split-step Fourier propagator is a one-way wave propagation method that has been widely used to simulate primary forward and backward (reflected) deterministic/random wave propagation due to its fast computational speed and limited computer memory requirement. The method is useful for rapid modeling of seismic-wave propagation in heterogeneous media where forward scattered waveforms can be considered to be dominant or reverberations can be ignored. The method is based on a solution to the one-way wave equation that requires expanding the square root of an operator and splitting of the resulting noncommutative operators to allow calculation by transferring wave fields between the space and wavenumber domains. Previous analysis of the accuracy of the method has focused on the error related to only a portion of the approximations involved in the propagator. To better understand the accuracy of the propagator, we present a complete formal and numerical accuracy analyses. Our formal analysis indicates that the dominant error of the propagator increases as the first order in the marching interval. We show that nonsymmetrically and symmetrically split-step marching solutions have the same first-order error term. Their second- and third-order error terms are similar. Therefore, the differences between the accuracy of different split-step marching solutions are insignificant. This conclusion is confirmed by our numerical tests. The relation among the phase error of the split-step Fourier propagator, relative velocity perturbation, and propagation angle is numerically studied. The results suggest that the propagator is accurate for up to a 60° propagation angle from the main propagation direction for media with small relative velocity perturbations (10%).