This textbook can be viewed as a "how-to" manual for solving quantum inverse problems, for deriving the potential from spectra or scattering data, and as a quantum "picture book" which aims to enhance the reader's quantum intuition. The formal exposition of inverse methods is paralleled by a discussion of the direct problem. Differential and finite-difference equations are presented side by side. The common features and advantages of a variety of solution methods are analyzed. To foster a better understanding, the physical meaning of the mathematical quantities are discussed. Wave confinement in continuum bound states, resonance and collective tunnelling, energy shifts and the spectral and phase equivalence of various interactions are some of the physical problems covered.