The existence of dark energy in the universe is hypothesized to explain the accelerated expansion of today’s universe and the inflationary expansion of very early universe; accelerated expansion of today’s universe is proved by observational data gathered from ground-based and space telescopes, from type Ia supernovas; and inflation of the early universe is assumed to solve issues of the Big Bang theory, for instance, flatness problem, and isotropy and homogeneity of the universe in large scale.There are different models of dark energy; the first and simplest one, is cosmological constant (or vaccum energy), that is constant all the time and space; the others are quintessential models which are dynamics, evolving and they can be inhomogeneous. In this thesis, two different quintessential models of dark energy, scalar field and non-metric quintessence, have been studied.Chapter one of this thesis is a brief review of general relativity and cosmology; in this chapter the need for a field which provides negative pressure and accelerates the expansion of our universe is discussed.Chapter two, is the study of the equations of motion of scalar field with exponential potential. In this chapter I’ve studied the equilibrium points of the phase plots found from the dynamics of dimensionless variables of the scalar field. Then I’ve solved the equations of the evolution of a universe, filled with scalar field and background fluids, numerically. Plots of the evolution of density parameters found from this model show that this model with exponential potential, is not consistent with our universe.In chapter three, equations of motion of a two-field system (scalar matter field with a power-law potential and graviscalar field), are derived from the lagrangian defining non-metric quintessence. I’ve found the phase plots of the system, and solved the equations of the evolution of a universe, filled with this two-field and background fluids (dark matter, baryonic matter and radiation), numerically. plots of the evolution of density parameters, are very compatible with the real picture of the history of the universe; and today’s values of density parameters and Hubble parameter, age of the universe and the size of particle horizon, found from this model, are very close to their measured values (and sometimes the same as measured values). Although this model is very compatible with our universe, it doesn’t explain inflation; and two-field system is in fact one scalar matter field.In order to explain the inflation of the early universe, I decided to define a hybrid potential of scalar matter field, that is discussed very briefly in chapter four. Hybrid potential is a combination of exponential (gaussian) and power-law potential; its exponential part dominates early universe and it can explain inflation, and power-law part of the hybrid potential is a good alternative for dark energy. The plots found from the numerical solution of the system of equations of dimensionless variables, are very compatible with the real history of our universe; and today’s values of density parameters of dark energy, dark matter, baryonic matter, and Hubble parameter, and the age and size of our universe, found from this model, are very close to (and some times the same as) measured values. I found more compatible answers assuming that radiation dominates our universe at a very early stage and later on, exponential potential dominates and causes rapid expansion of theearly universe.