This book presents an analytical theory on the electronic states in ideal low-dimensional systems and finite crystals, recently developed by the author, based on a differential equation theory approach. It gives some exact and general fundamental understandings on the electronic states in ideal low-dimensional systems and finite crystals, and provides new insights into some fundamental problems in low-dimensional systems such as the surface states, quantum confinement effects, etc., some of which are quite different from what is traditionally believed in the solid state physics community.