The book elucidates the physics of classical and quantum fluctuations in nonlinear oscillators and provides a unifying insight into fluctuation phenomena in a variety of vibrational systems of current interest. The considered systems are mesoscopis: they are small, so that fluctuations play an important role, but can be individually accessed. The book consists of chapters written by leading experts in the field. The chapters are self-contained and complement each other. They describe major types of nonlinear mesoscopic vibrational systems and the new quantum and classical physics learned using these systems. Also described are new approaches to quantum and classical measurements. The discussed topics include nonlinear dynamics, bistability, and quantum control of microwave cavity modes coupled to qubits; measurements with bifurcation amplifiers based on modulated vibrational systems and new types of such amplifiers; switching rate scaling and a new quantum mechanism of switching in modulated systems; nonlinear wave mixing, parametric excitation, and amplification in the quantum regime; collective phenomena in coupled modulated vibrational systems and the interaction-induced breaking of the time-translation symmetry; quantum back-action in strongly coupled electron-vibrational systems and the effect on the vibrations of the shot noise from spin current; new mechanisms of vibrational relaxation; and the quantum-classical correspondence in the strongly nonlinear regime. The specific systems considered in the book include Josephson junctions, microwave cavities containing qubits or other devices based on Josephson junctions, nano- and micro-mechanical systems, carbon nanotubes, cold atoms, and nano-magnetic oscillators.