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 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.