This book presents several solved problems and exercises related to vibrations in mechanical systems. Although the theory of mechanical vibrations is not dealt with in the book, a review of main mathematical equations needed to solve the proposed exercises is present in the Chapter 1 in order to make the book self-contained as much as possible. The structure of this book is such as to be intended as an exercise textbook for several courses on vibrations of mechanical systems for both Bachelor and Master of Science degrees and also as a reference for practicing engineers. The book begins with very elementary problems and raises the level gradually from linear single degree-of-freedom to complex non-linear multiple degrees-of-freedom examples. The problem of the linearization of the equation(s) of motion and of the computation of the linearized constraint forces is widely treated. Finally, for some exercises the numerical solution of linear and non-linear equations is presented and discussed.