A family of transport synthetic acceleration (TSA) methods for iteratively solving within-group scattering problems is presented. A single iteration in these schemes consists of a transport sweep followed by a low-order calculation, which itself is a simplified transport problem. The method for isotropic-scattering problems in X- Y geometry is described. Our Fourier analysis of a model problem for equations with no spatial discretization shows that a previously proposed TSA method is unstable in two dimensions but that our modifications make it stable and rapidly convergent. The same procedure for discretized transport equations, using the step characteristic and two bilinear discontinuous methods, shows that discretization enhances TSA performance. A conjugate gradient algorithm for the low-order problem is described, a crude quadrature set for the low-order problem is proposed, and the number of low-order iterations per high-order sweep is limited to a relatively small value. These features lead to simple and efficient improvements to the method. TSA is tested on a series of problems, and a set of parameters is proposed for which the method behaves especially well. TSA achieves a substantial reduction in computational cost over source iteration, regardless of discretization parameters or material properties, and this reduction increases with the difficulty of the problem.