The aim of the work described in this thesis is to extend the theory of Volterra functionals to new classes of nonlinear systems and to develop improved methods for applying- the existing theory to make it a more effective and useful method for nonlinear system analysis. A general theory of multidimensional Laplace transforms is developed to solve a class of multivariable nonlinear equations. Non-zero initial conditions are taken into consideration to make the solution completely general and suitable for application to practical engineering- problems. The theory is applied to analyse the transient response of a synchronous machine taking into consideration pole saliency and nonlinear damping. The convergence of Volterra functional series is studied using Barrett's method of majorant series- a series which dominates the original Volterra series term by term. The method is applied to determine a sufficient condition on the input reactivity function to ensure a bounded growth of neutrons in a temperature-feedback controlled nuclear reactor. The effects of the non-zero initial conditions on the reactor response are clearly shown. The limitations of this approach to study bounded-input, bounded-output stability of a nonlinear system are also discussed. The theory of Volterra functionals is extended to multi-input, multi-output nonlinear systems. It is used for the analysis of the transient as well as steady state response of a diode-ring multiplier circuit. Analytical deductions have been compared with the experimental results. Multilinear parametric transfer functions are introduced to analyse a class of time-varying nonlinear systems. The solution is obtained in the form of a Volterra series expansion. This new method of analysis is applied to a nonlinear control problem, and the results are compared with those obtained by other researchers in order to establish the superiority of this new method in achieving a faster convergence to the exact solution. A theory of multidimensional Mellin transforms is developed to analyse Euler-Cauch type time-varying nonlinear systems using Volterra functional expansion. The theory is applied to a physical control problem involving time-varying system parameters. The analysis gives the effects of the system parameters on the system response.