Electronic excitations are central to most measured spectra, but calculating excited states is more complex than ground-state calculations, which are efficiently handled by density-functional theory (DFT). Two main approaches for electronic excitations are many-body perturbation theory (MBPT) and time-dependent density-functional theory (TDDFT). MBPT uses Green's functions and the self-energy Σ, with the GW approximation being a key method. The Bethe-Salpeter equation describes electron-hole interactions. TDDFT, on the other hand, uses a density-based approach and a screening equation with a two-point interaction kernel. While TDDFT is practical, it has limitations in describing absorption spectra, leading to incorrect excitation energies and missing bound excitonic states. Improving TDDFT potentials and kernels is an active area of research. MBPT and TDDFT complement each other, with MBPT providing insights into quasiparticle properties and TDDFT offering a density-based framework. This review compares the theoretical and practical aspects of both approaches, highlighting their numerical implementations and current achievements. It discusses the importance of effective Hamiltonians, self-energy, and interactions in both methods, and addresses the challenges in accurately describing electronic excitations, particularly in solids. The review also covers the role of the dielectric function, local-field effects, and the connection between TDDFT and Bethe-Salpeter approaches. It emphasizes the need for improved approximations in TDDFT to better describe electronic excitations and highlights the significance of electron-hole interactions in determining spectral properties. The review concludes with an overview of the current state of research and open questions in the field.Electronic excitations are central to most measured spectra, but calculating excited states is more complex than ground-state calculations, which are efficiently handled by density-functional theory (DFT). Two main approaches for electronic excitations are many-body perturbation theory (MBPT) and time-dependent density-functional theory (TDDFT). MBPT uses Green's functions and the self-energy Σ, with the GW approximation being a key method. The Bethe-Salpeter equation describes electron-hole interactions. TDDFT, on the other hand, uses a density-based approach and a screening equation with a two-point interaction kernel. While TDDFT is practical, it has limitations in describing absorption spectra, leading to incorrect excitation energies and missing bound excitonic states. Improving TDDFT potentials and kernels is an active area of research. MBPT and TDDFT complement each other, with MBPT providing insights into quasiparticle properties and TDDFT offering a density-based framework. This review compares the theoretical and practical aspects of both approaches, highlighting their numerical implementations and current achievements. It discusses the importance of effective Hamiltonians, self-energy, and interactions in both methods, and addresses the challenges in accurately describing electronic excitations, particularly in solids. The review also covers the role of the dielectric function, local-field effects, and the connection between TDDFT and Bethe-Salpeter approaches. It emphasizes the need for improved approximations in TDDFT to better describe electronic excitations and highlights the significance of electron-hole interactions in determining spectral properties. The review concludes with an overview of the current state of research and open questions in the field.