This chapter serves as an introduction to the field of random vibrations, which in recent years has found extensive applications in structural dynamics, machine vibrations, aircraft vibrations, as well as in nondestructive testing evaluation and structural health monitoring. The first part of this work reviews the fundamental concepts underlying random vibrations. The presentation is by necessity brief, as there is extensive bibliography on this field [1, 2]. The second part examines the response of both single degree-of-freedom (SDOF) and multiple degree-of-freedom (DOF) structural systems to stochastic input, with both time and frequency domain techniques covered. In analyzing SDOF systems, basic structural dynamics is adequate [3], but MDOF representations of complex structural systems require use of numerical approaches such as the finite element method (FEM) [4] and the boundary element method (BEM) to a lesser extent. Finally, a numerical example serves to illustrate the concepts and methodologies presented herein.