In this dissertation, three different methods for solving the Boltzmann neutron transport equation (and its low-order approximations) are developed in general geometry and implemented in 1D slab geometry. The first method is for solving the fine-group diffusion equation by estimating the in-scattering and fission source terms with consistent coarse-group diffusion solutions iteratively. This is achieved by extending the subgroup decomposition method initially developed in neutron transport theory to diffusion theory. Additionally, a new stabilizing scheme for on-the-fly cross section re-condensation based on local fixed source calculations is developed in the subgroup decomposition framework. The method is derived in general geometry and tested in 1D benchmark problems characteristic of Boiling Water Reactors (BWR) and Gas Cooled Reactor (GCR). It is shown that the method reproduces the standard fine-group results with 3-4 times faster computational speed in the BWR test problem and 1.5 to 6 times faster computational speed in the GCR core. The second method is a hybrid diffusion transport method for accelerating multi-group eigenvalue transport problems. This method extends the subgroup decomposition method to efficiently couple a coarse-group high-order diffusion method with a set of fixed-source transport decomposition sweeps to obtain the fine-group transport solution. The advantages of this new high-order diffusion theory are its consistent transport closure, straight forward implementation and numerical stability. The method is analyzed for 1D BWR and High Temperature Test Reactor (HTTR) benchmark problems. It is shown that the method reproduces the fine-group transport solution with high accuracy while increasing the computationally efficiency up to 16 times in the BWR core and up to 3.3 times in the HTTR core compared to direct fine-group transport calculations. The third method is a new spatial homogenization method in transport theory that reproduces the heterogeneous solution by using conventional flux weighted homogenized cross sections. By introducing an additional source term via an “auxiliary cross section” the resulting homogeneous transport equation becomes consistent with the heterogeneous equation, enabling easy implementation into existing solution methods/codes. This new method utilizes on-the-fly re-homogenization, performed at the assembly level, to correct for core environment effects on the homogenized cross sections. The method is derived in general geometry and continuous energy, and implemented and tested in fine-group 1D slab geometries typical of BWR and GCR cores. The test problems include two single assembly and 4 core configurations. It is believed that the coupling of the two new methods, namely the hybrid method for treating the energy variable and the new spatial homogenization method in transport theory set the stage, as future work, for the development of a robust and practical method for highly efficient and accurate whole core transport calculations.