This review article by Matthias Brack provides an overview of the theoretical approaches used to describe simple metal clusters, focusing on the jellium model and semiclassical methods. The jellium model, which treats the ionic cores as a uniform positive background and the valence electrons as interacting in this background, has been widely successful in explaining the empirical properties of metal clusters, particularly alkali metals. The review covers the hierarchy of theoretical approximations leading to the jellium model, including the local-density approximation (LDA) for exchange and correlation effects, and the semiclassical approximation to the single-particle density matrix. The physical properties discussed include ground-state binding energies, ionization potentials, and dipole polarizabilities. The review also explores the collective electronic excitations from the perspective of cluster response, including sum rules. The jellium model is shown to be effective in describing the supershell structure in large alkali clusters, which cannot be explained by more structural models. The review highlights the strengths and limitations of the jellium model, emphasizing its simplicity and applicability to large clusters, while acknowledging its neglect of ionic structure. The article also discusses the connection between microscopic theories and classical limits, particularly in the context of large-N expansions and the liquid-drop model.This review article by Matthias Brack provides an overview of the theoretical approaches used to describe simple metal clusters, focusing on the jellium model and semiclassical methods. The jellium model, which treats the ionic cores as a uniform positive background and the valence electrons as interacting in this background, has been widely successful in explaining the empirical properties of metal clusters, particularly alkali metals. The review covers the hierarchy of theoretical approximations leading to the jellium model, including the local-density approximation (LDA) for exchange and correlation effects, and the semiclassical approximation to the single-particle density matrix. The physical properties discussed include ground-state binding energies, ionization potentials, and dipole polarizabilities. The review also explores the collective electronic excitations from the perspective of cluster response, including sum rules. The jellium model is shown to be effective in describing the supershell structure in large alkali clusters, which cannot be explained by more structural models. The review highlights the strengths and limitations of the jellium model, emphasizing its simplicity and applicability to large clusters, while acknowledging its neglect of ionic structure. The article also discusses the connection between microscopic theories and classical limits, particularly in the context of large-N expansions and the liquid-drop model.