The simple problem of a quantum mechanical particle moving freely inside a box is used as a paradigm to understand a number of basic aspects of quantum mechanics, including the roles played by boundary conditions, dimensionality, symmetry, degeneracy, and so on. Starting from the simple one-dimensional cases of a line interval and a ring, we progress to integrable and nonintegrable billiard problems in two dimensions. The energy spectrum of a free particle on a 2-sphere is discussed. Finally, we touch upon the inverse problem of deducing the geometrical information carried by the energy spectrum of the particle via the Weyl and trace formulas.