Laser wakefield acceleration of electrons holds great promise for producing ultra-compact stages of GeV scale, high quality electron beams for applications such as x-ray free electron lasers and high energy colliders. Ultra-high intensity laser pulses can be self-guided by relativistic plasma waves over tens of vacuum diffraction lengths, to give >1 GeV energy in cm-scale low density plasma using ionization-induced injection to inject charge into the wake at low densities. This thesis describes a series of experiments which investigates the physics of LWFA in the self-guided blowout regime. Beginning with high density gas jet experiments the scaling of the LWFA-produced electron beam energy with plasma electron density is found to be in excellent agreement with both phenomenological theory and with 3-D PIC simulations. It is also determined that self-trapping of background electrons into the wake exhibits a threshold as a function of the electron density, and at the densities required to produce electron beams with energies exceeding 1 GeV a different mechanism is required to trap charge into low density wakes. By introducing small concentrations of high-Z gas to the nominal He background the ionization-induced injection mechanism is enabled. Electron trapping is observed at densities as low as 1.3 x …