Linear vibrations is a topic of central interest in numerous fields of engineering and is widely applicable to civil, mechanical, aerospace, and biomedical engineering. This book deals with the elements of the theory of linear vibrations and its applications. The theory of linear vibrations is systematically developed starting from the basics. In a systematic manner, the author deals with single-degree of freedom systems before progressing to the more complex multi-degree of freedom systems. Topics such as Laplace transforms, complex frequency responses, elements of feedback control and aspects of time-delayed control and its applications are discussed at the end. A set of graded problems, to exercise the understanding of the student, is provided at the end of each chapter. Numerous illustrative examples are provided in each chapter alongside the development of the theory. In addition, there are two chapters specifically devoted to illustrative examples and applications.By including topics from stability and control theory, the book aims to provide the student with a broader view of the application areas to which the theory is germane. It is meant to be a text for a one semester, first course, on the subject, suitable for seniors and/or first year graduate students in engineering. Applied mathematicians will also find this book useful.