This paper summarizes the formulation and numerical implementation of a spectral scheme specially designed for dynamic anti-plane shear (mode III) fracture problems. The scheme allows for a wide variety of simulations ranging from the dynamic loading of stationary cracks to the spontaneous propagation of faults. The method is based on a spectral form of the elastodynamic relation between the shearing tractions acting on the fracture surface and the resulting slip velocity response for a planar two-dimensional crack in an infinite linearly elastic medium. The formulation is expressed in the Fourier domain and involves a convolution over the past slip or slip rate history. Conversion between spectral and real domains is performed through the fast Fourier transform algorithm. The time-integration scheme is explicit and a variety of constitutive laws can be used to express the strength of the material on the fault plane. The stability and accuracy of the numerical scheme are discussed through comparison with existing analytical solutions involving non-propagating and propagating cracks. The extraction of the dynamic stress intensity factor from the computed slip history is described.