The article reviews the theoretical and practical aspects of two prominent approaches for describing electronic excitations: many-body perturbation theory and time-dependent density-functional theory (TDDFT). Many-body perturbation theory, based on Green's-function equations, considers the electron-hole Green's function and the electron's self-energy Σ, with Hedin's GW approach providing a good approximation for Σ. TDDFT, on the other hand, offers a practical advantage by depending on density rather than multivariable Green's functions, leading to a screening equation similar to the Bethe-Salpeter equation. The review compares these approaches, focusing on their numerical implementations and applications to spectroscopic measurements such as photoemission and absorption. It highlights the limitations of the simple adiabatic local-density approximation in TDDFT and the need for improved potentials and kernels. The article also discusses the challenges and progress in calculating electronic excitations, emphasizing the importance of accurate descriptions of spectroscopic experiments.The article reviews the theoretical and practical aspects of two prominent approaches for describing electronic excitations: many-body perturbation theory and time-dependent density-functional theory (TDDFT). Many-body perturbation theory, based on Green's-function equations, considers the electron-hole Green's function and the electron's self-energy Σ, with Hedin's GW approach providing a good approximation for Σ. TDDFT, on the other hand, offers a practical advantage by depending on density rather than multivariable Green's functions, leading to a screening equation similar to the Bethe-Salpeter equation. The review compares these approaches, focusing on their numerical implementations and applications to spectroscopic measurements such as photoemission and absorption. It highlights the limitations of the simple adiabatic local-density approximation in TDDFT and the need for improved potentials and kernels. The article also discusses the challenges and progress in calculating electronic excitations, emphasizing the importance of accurate descriptions of spectroscopic experiments.