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Observational evidence for the accelerated expansion of the universe requiresdark energy for its explanation if general relativity is an accurate model ofgravity. However, dark energy is a mysterious quantity and we do not know muchabout its nature so understanding dark energy is an exciting scientificchallenge. Cosmological dark energy models are fairly well tested in the lowand high redshift parts of the universe. The highest of the low redshift,$z\sim2.3$, region is probed by baryon acoustic oscillation (BAO) measurementsand the only high redshift probe is the cosmic microwave background anisotropywhich probes the $z\sim1100$ part of redshift space. In the intermediateredshift range $2.3 < z < 1100$ there are only a handful of observationalprobes and cosmological models are poorly tested in this region.  In this thesis we constrain three pairs of general relativistic cosmologicaldark energy models using observational data which reach beyond the current BAOlimit. We use quasar X-ray and UV flux measurements, the current version ofthese data span $0.009 \leq z \leq 7.5413$. We have discovered that most ofthese data cannot be standardized using the proposed method. However, the lowerredshift part, $z \lesssim 1.5-1.7$, of these data are standardizable and canbe used to derive lower-$z$ cosmological constraints. Another data set we useare gamma-ray burst measurements which span $0.3399 \leq z \leq 8.2$.Cosmological constraints derived from these data are significantly weaker than,but consistent with, those obtained from better-established cosmologicalprobes. We also study and standardize 78 reverberation-measured Mg II time-lagquasars in the redshift range $0.0033 \leq z \leq 1.89$ by using theirradius-luminosity relation. We also study 118 reverberation-measured H$\beta$time-lag quasars which span $0.0023 \leq z \leq 0.89$.

Using quasar and gamma-ray burst measurements to constrain cosmological dark energy models