In recent years, with the improvement of accuracy, density functional theory (DFT) has become a useful tool in the investigation of the electronic structures of atoms, molecules, and solids, due to its high computational efficiency, compared to the traditional wavefunction theory. In this theory, only the exchange-correlation energy as a functional of the electron density has to be approximated. While many density functional approximations for exchange and correlation have been proposed, a chemically accurate and yet universal exchange-correlation functional is still desired. To construct or improve an exchange-correlation functional, the investigation of the existent functionals is helpful. For example, comparison of the approximate exchange-correlation energy density with the exact one is useful for us to identify regions of space in which the approximation works or fails. Unfortunately, the exchange-correlation energy density is not uniquely defined. Though there have existed several definitions for this quantity, the conventionally-defined one is of special importance in DFT, since they are related to the exchange-correlation hole. In this study, (1) we investigated the most fundamental level of the widely-used Colle-Salvetti correlation in chemistry for the uniform electron gas and found that it gave only 25% of the exact correlation energy and not 100% as previously believed in the literature. (2) we derived the asymptotic behavior of the exchange energy density near a nucleus and then built this and other correct behaviors into a new density functional for the exchange energy. (3) we proposed an accurate MGGA-based hydrid exchange-correlation functional by mixing some exact exchange in this functional. This hybrid functional improves or competes with the previously established hybrid functionals in the literature. (4) we proposed a sophisticated hybrid model which predicts the conventional correlation energy density from a correlated wavefunction. This model allows us to compare various density functional approximations for the correlation energy density with the exact conventional correlation energy density in simple systems. (5) starting with the second-order gradient expansion for the exchange hole without integration by parts, we performed the real-space cutoff procedure. The resultant nonempirical EMGGA0x is used to construct the controlled-interpolation EMGGA2x between the slowly-varying density and the iso-orbital region