This book presents an exploration of the wave and vibration equation in one, two and three dimensions, with emphasis on singular solutions. The distinction between the wave treatment and the vibration treatment is particularly discussed with the causality principle being the leading principle for waves in this context. The necessity of regularization of the singular solutions is presented whilst the scattered waves are differentiated from the reflected (and refracted) waves, according to Huygens principle. The physical content of the wave equation is underlined. Relevant applications are included and some more exotic phenomena are discussed, such as pulses, tsunami and storm breakers, the ringing of bells and the collapsing of towers, and classical waves and vibrations in an elastic half-space or a sphere. This book is oriented to students, instructors, teachers, researchers in physics and applied mathematics, as well as engineers and other practitioners of mathematical physics.