This dissertation documents the development of a broadband electron spectrometer (ESM) for GeV class Laser Wakefield Accelerators (LWFA), the production of high quality GeV electron beams (e-beams) for the first time in a LWFA by using a capillary discharge guide (CDG), and a statistical analysis of CDG-LWFAs. An ESM specialized for CDG-LWFAs with an unprecedented wide momentum acceptance, from 0.01 to 1.1 GeV in a single shot, has been developed. Simultaneous measurement of e-beam spectra and output laser properties as well as a large angular acceptance (> {+-} 10 mrad) were realized by employing a slitless scheme. A scintillating screen (LANEX Fast back, LANEX-FB)--camera system allowed faster than 1 Hz operation and evaluation of the spatial properties of e-beams. The design provided sufficient resolution for the whole range of the ESM (below 5% for beams with 2 mrad divergence). The calibration between light yield from LANEX-FB and total charge, and a study on the electron energy dependence (0.071 to 1.23 GeV) of LANEX-FB were performed at the Advanced light source (ALS), Lawrence Berkeley National Laboratory (LBNL). Using this calibration data, the developed ESM provided a charge measurement as well. The production of high quality electron beams up to 1 GeV more » from a centimeter-scale accelerator was demonstrated. The experiment used a 310 {micro}m diameter gas-filled capillary discharge waveguide that channeled relativistically-intense laser pulses (42 TW, 4.5 x 10{sup 18} W/cm{sup 2}) over 3.3 centimeters of sufficiently low density ({approx_equal} 4.3 x 10{sup 18}/cm{sup 3}) plasma. Also demonstrated was stable self-injection and acceleration at a beam energy of {approx_equal} 0.5 GeV by using a 225 {micro}m diameter capillary. Relativistically-intense laser pulses (12 TW, 1.3 x 10{sup 18}W/cm{sup 2}) were guided over 3.3 centimeters of low density ({approx_equal} 3.5 x 10{sup 18}/cm{sup 3}) plasma in this experiment. A statistical analysis of the CDG-LWFAs performance was carried out. By taking advantage of the high repetition rate experimental system, several thousands of shots were taken in a broad range of the laser and plasma parameters. An analysis program was developed to sort and select the data by specified parameters, and then to evaluate performance statistically. The analysis suggested that the generation of GeV-level beams comes from a highly unstable and regime. By having the plasma density slightly above the threshold density for self injection, (1) the longest dephasing length possible was provided, which led to the generation of high energy e-beams, and (2) the number of electrons injected into the wakefield was kept small, which led to the generation of high quality (low energy spread) e-beams by minimizing the beam loading effect on the wake. The analysis of the stable half-GeV beam regime showed the requirements for stable self injection and acceleration. A small change of discharge delay t{sub dsc}, and input energy E{sub in}, significantly affected performance. The statistical analysis provided information for future optimization, and suggested possible schemes for improvement of the stability and higher quality beam generation. A CDG-LWFA is envisioned as a construction block for the next generation accelerator, enabling significant cost and size reductions. « less