The method of characteristics (MOC) is a deterministic transport method that has been applied to large-scale problems including those in reactor physics and radiative transfer. Long characteristics, (LC) methods, have been used extensively to discretize and solve transport problems in the spatial domain. There is a need for an equally adequate time-dependent discretization for these transport problems. The new contributions from this research include the development of a space-time long characteristic (STLC) method with various source approximations including several that employ a piece-wise linear (PWL) approximation spatially. In the prism-PWL (PPWL) method the coefficient of each PWL spatial function is linear in time in each space-time cell. Along with STLC, a PWL-LC method is developed for steady-state problems in (x, y) and (x, y, z). The methods developed in this work use least-squares projections to determine the coefficients of their source approximations. This work presents a detailed asymptotic analysis of the PWL-LC and STLC methods in the thick diffusion limit, which is of special interest in radiative transfer problems. This is the first such analysis reported for LC methods and these new methods are the first that are expected to perform well in this limit. Results from test problems executed with a modified version of the Parallel Deterministic Transport code, PDT, show the PWL-LC and STLC methods are more accurate than current methods for streaming problems. From asymptotic analysis and test problems, it is found that the steady-state PWL-LC method is accurate in the thick diffusion limit with solutions similar to those of analogous discontinuous finite element method, DFEM, solutions. Similarly, the PPWL-STLC method is found to be accurate in time-dependent thick diffusive problems. STLC is also a promising method for massively parallel applications because it permits the use of track-based sweeping, which appears to have significant advantages over cell-based sweeping. This is a key topic recommended for further research.