Recent advances in the development of high power short pulse laser systems has opened a new regime of laser plasma interactions for study. The thesis is presented in two parts. In Part I, we consider the implications of these high power laser pulses for the interaction with a uniform underdense plasma, with particular regard to plasma-based accelerators. We present a scheme for the resonant excitation of large electrostatic Wakefields in these plasmas using a train of ultra-intense laser pulses. We also present an analysis of the resonant mechanism of this excitation based on consideration of phase space trajectories. In Part II, we consider the transition from linear Resonance Absorption to nonlinear absorption processes in a linear electron density profile as the intensity of the incident radiation increases and the scale length of the density profile decreases. We find that the electron motion excited by an electrostatic field exhibits some extremely complicated dynamics with bifurcations to period doubling and chaotic motion as the strength of the driving field is increased or the density scale length is decreased. We also present some results obtained from particle simulations of these interactions.