  Recent observations opened up a new window on the inflationary model
building. As it was firstly reported by the WMAP data, there may be some
indications of statistical anisotropy on the CMB map, although the statistical
significance of these findings are under debate. Motivated by these
observations, people begun considering new inflationary models which may lead
to statistical anisotropy. The simplest possible way to construct anisotropic
inflation is to introduce vector fields. During the course of this thesis, we
study models of anisotropic inflation and their observational implications such
as power spectrum, bispectrum etc. Firstly we build a new model, which contains
the gauge field which breaks the conformal invariance while preserving the
gauge invariance. We show that in these kind of models, there can be an
attractor phase in the evolution of the system when the back-reaction of the
gauge field becomes important in the evolution of the inflaton field. We then
study the cosmological perturbation theory in these kind of models. More
specifically, we calculate the anisotropic corrections due to the presence of
the vector field. We then generalize the separate universe formalism to our
anisotropic set up and use it in some specific examples of anisotropic
inflation. Finally, we connect the primordial anisotropies to the specific
examples and to CMB observations. We calculate the TT, TE, TB, EB and BB
correlation in the model of charged scalar field model and look for the unique
signatures that the anisotropic inflation can have on the CMB map. Any future
detection of these statistical anisotropies would rule out the isotropic FRW
models.
