This two-volume set provides a comprehensive study of fractional random vibration from the perspective of theory and practice. Volume I deals succinctly with the theories of fractional processes and fractional vibration systems. A major focus of fractional vibrations is the derivation of analytical expressions for the frequency transfer functions of seven classes of fractional vibrations using elementary functions. This is considered from the perspective of the functional form of linear vibrations with frequency-dependent mass, damping, or stiffness. The present results serve as a basis for the study of the novel and frontier topic of fractional processes passing through fractional vibration systems, which is discussed in Volume II. The title will be essential reading for students, mathematicians, physicists, and engineers interested in fractional random vibration phenomena.