Spectral and pseudospectral methodr for spatially discretiring the first-order transport equation in slab geometry are discussed and compared with finite element and other discretization methods. Spectral methods are shown to have exponential (“infinite order”) convergence for homogeneous problems, enabling them to achieve extremely accurate solutions at lower computational costs than other methods. A “spectral element” method is proposed which extends this exponential convergence to heterogeneous problems. Numerical results are given supporting all theoretical claims, and applicability of the method to physically significant problems is discussed. We conclude that the spectral element method is probably useful mainly for high-accuracy calculations of one-dimensional problems; in higher dimensions the transport solution lacks the smoothness which for realistic problems would allow the method to achieve exponential convergence.