Several applications of the degenerate kernel technique (DKT) for treating the speed dependence in steady-state neutron thermalization calculations are studied both analytically and computationally. An iterative improvement technique is developed for fine thermal spectrum calculations. It is shown that the size of the degenerate kernel expansion (DKE) required to obtain consistent accuracy with a given number of discrete speed mesh points can be decoupled from the speed mesh structure by such a technique. This decoupling allows a more efficient numerical solution and hence a savings in computation time. The solution of the integral transport equation within the isotropic scattering approximation is also studied within the DKT framework. The DKT formalism allows a considerable reduction in the dimensionality of the numerical representation of this problem, hence implying reduced computation costs. Finally, the DKE has been employed within the invariant-imbedding transport formalism to calculate the reflection (R) and transmission (T) probabilities for thermal neutrons incident upon a slab. Once again the DKT leads to a very considerable reduction in computation time and storage when compared with multigroup approaches. Numerical methods for solving the invariant imbedding-DKT equations for R and T have been developed and computationally verified as both accurate and efficient.