This chapter analyzes the free and forced vibrations of systems with multiple degrees of freedom. It covers both damped and undamped free vibrations, introducing key concepts such as natural frequencies, mode shapes, and complex modes. For forced vibrations, the frequency response function matrix is used to discuss the system response to varying excitation frequencies. The chapter concludes with an overview of modal analysis. The physics of free and forced vibrations are initially illustrated with examples, followed by the general theory. Matlab codes are provided to solve generalized and quadratic eigenproblems.