This two-volume set provides a comprehensive study of fractional random vibration from the perspective of theory and practice. By fractional random vibration, we mean the random vibration of fractional vibration systems driven by fractional processes. Volume I explores the fundamentals of fractional processes and fractional vibration systems. A major focus is the derivation of analytical expressions for the frequency transfer functions of seven classes of fractional vibrations using elementary functions. Theoretical support is provided by the functional form of linear vibrations with frequency-dependent mass, damping, or stiffness. Volume II provides the analytical expressions for the responses of seven classes of fractional vibration systems subjected to fractional Gaussian noise, generalized fractional Gaussian noise, fractional Brownian motion, the fractional Ornstein-Uhlenbeck process, the process with the von Kármán spectrum, and the one with the Pierson-Moskowitz spectrum.