<p>Density functional theory is a formally exact theory to describe ground state properties due to the existence of the exact functional. In practice, the usefulness of density functional theory relies on the accuracy of density functional approximations. After decades of effort of functional developments, the present-day state-of-the-art density functional approximations have achieved reasonably good accuracy for small systems. However, the error grows with system size. One of the dominant errors intrinsic in the mainstream density functional approximations is the delocalization error, which arises because of the violation of Perdew-Parr-Levy-Balduz (PPLB) linearity condition. The PPLB condition governs the formulation of the density functional theory for fractional-charge systems, for which the ground state energy for the exact functional, as a function of the fractional electron number, should yield a series of line-segments across the integer points. In this dissertation, by imposing the PPLB condition in a local, size-consistent way, we develop the local scaling correction (LSC) and its updated version, the localized orbital scaling correction (LOSC), which largely improve upon the mainstream density functional approximations across system sizes. With the LOSC, we open a door towards a systematic elimination of delocalization error. Besides the ground state functional development, we also develop a gentlest ascent dynamics approach for accessing the excited states via time-independent ground state density functionals. This is also useful for exploring Kohn-Sham energy landscapes of approximate density functionals. I will also review the PPLB formulation of density functional theory for fractionally charged systems, and show that it is equivalent to the formulation normally used for fractional system calculations under certain assumptions. Furthermore, I will examine the behavior of the fractional system energy as a function of the fractional number of electrons for different mainstream functionals, and relate it to their errors for integer systems.</p> / Dissertation