The objective of this chapter is to introduce the form of the equations of motion for systems with multiple degrees of freedom. This chapter first addresses these equations through two classical examples from mechanics. Next, the general form of the equations is derived in a broader context using the generalized Taylor series, highlighting the limitations and applicability of linear vibration theory. The chapter concludes with three solved examples that effectively illustrate the application of the main concepts developed herein. Matlab codes are provided throughout to aid in visualizing the physics of the problems studied.