Oscillations are ubiquitous in physics. For instance, in mechanics, we encounter them in compressed springs, pendula and elastic solids. Small oscillations give us a window into the types of phenomena that can occur in more general oscillations and also a first approximation to describing them. In this chapter, we discuss small oscillations (or vibrations) of a system with one degree of freedom around a static stable equilibrium. This includes free simple harmonic oscillations, damped oscillations, forced or driven oscillations as well as parametric resonance. Some extensions to nonlinear oscillations and to systems with more degrees of freedom and oscillations around time-dependent states are discussed in Chaps.  7 , 13 and 15 .