Seeking and developing an efficient and robust method for the accurate description of the band gaps of periodic solids continues to be a formidable challenge. Kohn-Sham density functional theory in the LDA or GGA always underestimates the band gaps due to the so-called “self-interaction error (SIE)”. To reduce the SIE and improve the band gap description, hybrid functionals and the state-of-the-art GW method are widely used. In this talk, I will focus on the recently proposed nonempirical dielectric-dependent range-separated hybrid functionals (CAM/DSH) and our recently developed cubic-scaling GW method and its extensions: renormalized G_RW₀ method and fully self-consistent GW (scGW) method. We identified the connections between different hybrid functionals and assessed the performances of the different methods. We found that single-shot CAM0 and DSH0 outperform conventional hybrid functionals and even the single-shot G0W0 method in the description of band gaps for a large variety of semiconductors and insulators. Our proposed renormalized G_RW₀ method, which renormalizes the Green’s function by using the one-shot eigenvalues/eigenfunctions of the CAM0/DSH0 Hamiltonian evaluated at the PBE density matrix, while the screened interaction is still evaluated using PBE one-electron energies and orbitals, much improves upon the conventional G₀W₀ method and are comparable with the eigenvalue self-consistent GW0 (evGW0) method, but with less computational efforts. As expected, without vertex corrections, the full scGW method yields unsatisfactory results. Nevertheless, it paves the way to further improve, for instance, by including the simplest possible vertex W.