With the aim of stating the fundamental principles and relationships of structural and mechanical vibrations, this guide focuses on the determination of response levels for dynamical systems excited by forces that can be modeled as stochastic processes. It concentrates material in the beginning of the text, with introductions to the fundamentals of stochastic modeling and vibration problems to acquaint students with applications. There are discussions on progressive topics which are the subject of ongoing research, including state-space analysis, nonlinear dynamics, and fatigue damage; the time history implications of bandwidth, with situations varying from narrowband to white noise; time domain integration techniques which provide viable alternatives to the calculus of residues; and an emphasis on time domain interpretations throughout. It includes a number of worked examples to illustrate the modelling of physical problems as well as the proper application of theoretical solutions.