Our Universe has an exciting history of accelerated expansion. Following its inception in an event known as the big bang, the Universe underwent a phase of exponential expansion called inflation. Although precise observations of the cosmic microwave background (CMB) radiation and large-scale structure support the inflationary paradigm, the absence of a firm physical mechanism for inflation has led to a plethora of theoretical embarkments attempting to understand its genesis. Inflation lasted only for a fraction of a second, but observations suggest that thirteen billion years after inflation, the Universe began accelerating once again. This acceleration, which continues till date, is attributed to a mysterious component, dubbed dark energy, that fills up our Universe and accounts for almost 73% of the total energy density in the Universe. In this thesis we study and develop models of inflation and dark energy, in the light of current observations. We begin, in Chapter 1, with a detailed introduction to cosmic acceleration, and discuss various models of inflation and dark energy that have been studied in the literature in the recent years. In Chapter 2, we discuss how a hierarchy of Hubble flow parameters, extended to include the evolution of the inflationary sound speed, can be applied to compare a general, single-field inflationary action with cosmological observational data. In Chapter 3, we study the six-field dynamics of D3-brane inflation for a general scalar potential on the conifold, finding simple, universal behavior, such as a power law dependence for the probability of Ne e-folds of inflation. Subsequent chapters study modifications to the theory of gravity in an attempt to understand dark energy. In Chapter 4, we establish the dynamical attractor behavior in scalar-tensor theories of gravity, providing a powerful framework to analyze such theories, predicting common evolutionary characteristics that can be compared against cosmological constraints. Chapter 5 develops a cascading cosmology framework in order to study the cosmological implications of a sixdimensional (6D) theory of gravity. We find that cascading cosmology can indeed lead to an accelerating Universe. Finally, in Chapter 6, we study the issue of ghostlike instabilities in the cascading framework, and propose a mechanism to obtain flat brane and bulk solutions. We conclude in Chapter 7 with a discussion of the main results presented in this thesis.