This thesis addresses the problem of finding numerical solutions to ordinary and algebraic differential equation systems. Our primary focus is the application of onestep numerical schemes to these problem classes. Firstly we concentrate on the narrower class of explicit Ordinary Differential Equation (O DE) systems. We analyse the theory necessary develop efficient algorithms based on our chosen one-step numerical schemes. These algorithms are then applied to the solution of a standard test set of ODE systems. The results are then compared with those obtained using standard software packages on the same problem test set. Our theory is then extended to include the wider class of Algebraic Differential Equation (more commonly called Differential Algebraic Equation (D A E ) ) systems. Based on this theory, we are able to adapt our one-step schemes to solve this harder class of problem. Once again the resulting algorithms are tested on a selection of problems and results are compared with those obtained from standard software packages. On all problems considered, we demonstrate that our techniques can often provide efficient alternatives to the more complex methods adopted in the standard software packages designed for these problem classes.