The objective of this chapter is to investigate the free vibratory response of systems with one degree of freedom. The presentation starts with undamped systems, and key concepts are presented using a practical demonstration that can be easily replicated. Subsequently, the analysis extends to cases where damping is present and the effect of different levels of damping on the system response is examined. Several examples are provided to illustrate the applicability of the theory. Some of them involve modeling, analysis, and validation of the theory by means of simple experiments that readers can carry out on their own.