The meta-GGA functionals and random phase approximation are tested for phase transitions and a strongly correlated transition metal oxide in this dissertation. One of the latest meta-GGA functionals is also employed to study the van der Waals bound system in surface science. Our main purpose is to reveal the performance of new exchange-correlation functionals on various properties and systems. We are also interested in seeking the possible relationship between the performance of a semilocal functional and its exchange enhancement factor. We have studied the structural phase transitions in crystalline Si (insulator to metal), SiO2 (insulator to insulator) and Zr (metal to metal) systems, as a test of exchange energy semilocal functionals on Jacob's ladder. Our results confirm the energy-geometry dilemma of GGAs in three systems. The most sophisticated non-empirical meta-generalized gradient approximations (meta-GGAs) such as TPSS (Tao-Perdew-Staroveov-Scuseria) and revTPSS (revised TPSS) give better lattice constants than PBE, but the phase transition parameters (energy difference and transition pressure) are smaller and less realistic than those from the latter GGA. However, the recent functionals of meta-GGA made simple family (MGGA_MS) behave differently to those previous meta-GGAs, predicting larger and more realistic phase transition parameters. Meanwhile, MGGA_MS also delivers the equilibrium geometry of crystalline materials similar to previous non-empirical meta-GGAs. In contrast to semilocal functionals, the nonlocal functionals such as the range-separated hybrid functional HSE06 (Heyd-Scuseria-Ernzerhof) and non-self consistent random phase approximation (RPA) are not only able to give the accurate equilibrium geometry , but also predict the realistic phase transition parameters for Si and SiO2 systems. The ground state of rutile-type vanadium dioxide (R-VO2) represents a great challenge to the current density functional theory. In this dissertation, we investigated the electronic structures and magnetism of R-VO2 using exchange-correlation functionals of all five rungs on Jacob's ladder. Our calculations show that all semilocal functionals (LSDA, GGAs and meta-GGAs) and hybrid functionals (HSE06) stabilize the spin-polarized states (ferromagnetic and anti-ferromagnetic states) over non-magnetic state, which are completely opposite to experimental observation. Surprisingly, LSDA gives the best energetic descriptions for magnetic and non-magnetic phases of R-VO2 among semilocal functionals and HSE06. Otherwise, RPA calculations are highly dependent on the inputs in the spin polarized case. With PBE inputs, RPA also fails, giving lower energies for spin-polarized states than for the non-magnetic phase. Meanwhile, the results are reversed using LSDA inputs. From the computed equilibrium cell volume, we observe the error cancellation in the exchange-correlation hole of most semilocal functionals in the spin-polarized calculations. LSDA and RPA do not fit to this picture. By analyzing the local magnetic moments of vanadium atoms, it is found that the magnetic property predicted from meta-GGA can be related to its exchange enhancement factor. The physisorption of a molecule on a transition metal surface is also another difficult problem in DFT because of the long-range van der Waals interactions. The recently developed MGGA_MS family of density functionals is able to capture a portion of intermediate range dispersion interactions. Therefore, we employed MGGA_MS2 to study the physisorption of CO2 on Pt (111) surface, and the results are compared to those of PBE, PBE+D2 and optB88-vdW methods. The computed binding curves confirm that that MGGA_MS2 indeed captures the van der Waals interactions near the equilibrium binding distance, and the obtained binding distance is also in good agreement with PBE+D2 and optB88-vdW calculations. By computing the electron density difference map (EDDM), we find that the electron densities of CO2 and Pt (111) surface are strongly polarized in optB88-vdW, creating the dipole moments in two subsystems. Such effect is reduced in MGGA_MS2. For PBE, the polarization of electron density is very weak, but not negligible. The α dependence in the exchange enhancement factor of a meta-GGA is the key to capture the intermediate range van der Waals interactions. In summary, a meta-GGA functional can step out of the famous "energy-geometry dilemma" , predicting good lattice constants and phase transition parameters at the same time. With the proper construction, a meta-GGA can even capture a portion of van der Waals interactions. The RPA is usually more accurate than semilocal functionals for many ground state properties. The strongly correlated systems like R-VO2 are still a big challenge to present-day density functional theory. We will continue to seek more accurate exchange-correlation functionals. / Physics