A semi-analytical technique for the solution of neutron and gamma-ray transport in one-dimensional finite systems is developed. The method is applicable to multivelocity, multiregion systems with arbitrary degree of anisotropy. The transport equation is written in the form of coupled integral equations separating the spatial and energy-angular transmissions. Legendre polynomial approximation in the direction cosine, and discrete ordinate representation in energy and spatial domain are used for radiation source and flux. The space and energy-angle transmission kernels are evaluated analytically and the integral equations are then solved by a fast-converging iterative technique. For a plane parallel beam of radiation incident on a slab, the virgin and the first collision flux are not amenable to polynomial expansion due to the singularities. For such a case, up to second collision, source is computed analytically and then recourse is taken to polynomial approximation. The computer code ASFIT written on the basis of the above formulation is briefly described. Convergence studies with the polynomial approximation, energy and spatial mesh width are described.