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 …