  The expansion rate of the Universe changes with time, initially slowing
(decelerating) when the universe was matter dominated, because of the mutual
gravitational attraction of all the matter in it, and more recently speeding up
(accelerating). A number of cosmological observations now strongly support the
idea that the Universe is spatially flat (provided the dark energy density is
at least approximately time independent) and is currently undergoing an
accelerated cosmological expansion. A majority of cosmologists consider "dark
energy" to be the cause of this observed accelerated cosmological expansion.
  The "standard" model of cosmology is the spatially-flat {\Lambda}CDM model.
Although most predictions of the {\Lambda}CDM model are reasonably consistent
with measurements, the {\Lambda}CDM model has some curious features. To
overcome these difficulties, different Dark Energy models have been proposed.
Two of these models, the XCDM parametrization and the slow rolling scalar field
model $\phi$CDM, along with "standard" {\Lambda}CDM, with the generalization of
XCDM and $\phi$CDM in non-flat spatial geometries are considered here and
observational data are used to constrain their parameter sets.
  In this thesis, we start with a overview of the general theory of relativity,
Friedmann's equations, and distance measures in cosmology. In the following
chapters we explain how we can constrain the three above mentioned cosmological
models using three data sets: measurements of the Hubble parameter $H(z)$,
Supernova apparent magnitudes, and the BAO peak length scale, as functions of
redshift $z$. We then discuss constraints on the deceleration-acceleration
transition redshift $z_{\rm da}$ using unbinned and binned $H(z)$ data.
  Finally, we incorporate the spatial curvature in the XCDM and $\phi$CDM model
and determine observational constraints on the parameters of these expanded
models.
