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