In last fifty years, Density-functional Theory (DFT) enjoys considerable success explaining the ground-state properties of bulk solids, surface, and even nano materials. However, this great success of DFT will be diminished once we go beyond the ground-state properties. An effective independent-particle picture is no more adequate for the many-electron systems in excited-state regime. Therefore, how to mimic properly the many-body effects in realistic electronic structures is the crucial work for developing schemes next to DFT . Based on the framework of many-body perturbation theory (MBPT), two main issues in many-electron interacting systems, quasiparticle and excited electron-hole pair effects, can be addressed by recently developed GW approximation (GWA) and solutions of the Bethe-Salpeter equation (BSE), respectively. On the other hand, the time-dependent density-functional theory (TDDFT) is an alternative approach to calculating electronic excitations which basically offers the important practical advantage of dependence on density rather than on multivariable Green's functions. In this minischool, we will include the fundamental theories, algorithms, practical aspects of ab-initio excitation calculations, and also their applications to the optical excitations and excitonic physics of novel materials.