Part I of this thesis is concerned with cavity optomechanical systems subject to periodic driving. We develop a Floquet approach to solve time-periodic quantum Langevin equations in the steady state, show that two-time correlation functions of system operators can be expanded in a Fourier series, and derive a generalized Wiener-Khinchin theorem that relates the Fourier transform of the autocorrelator to the noise spectrum. Weapply our framework to optomechanical systems driven with two tones. In a setting used to prepare mechanical resonators in quantum squeezed states, we nd and study the general solution in the rotating-wave approximation. In the following chapter, we show that our technique reveals an exact analytical solution of the explicitly time-periodic quantum Langevin equation describing the dual-tone backaction-evading measurement of a single mechanical oscillator quadrature due to Braginsky, Vorontsov, and Thorne [Science 209, 547 (1980)] beyond the commonly used rotating-wave approximation and show that our solution can be generalized to a wide class of systems, including to dissipatively or parametrically squeezed oscillators, as well as recent two-mode backaction-evading measurements. In Part II, we study nonreciprocal optomechanical systems with several optical and mechanical modes. We show that an optomechanical plaquette with two cavity modes coupled to two mechanical modes is a versatile system in which isolators, quantum-limited phase-preserving, and phase-sensitive directional ampliers for microwave signals can be realized. We discuss the noise added by such devices, and derive isolation bandwidth, gain bandwidth, and gain-bandwidth product, paving the way toward exible, integrated nonreciprocal microwave ampliers. Finally, we show that similar techniques can be exploited for current rectication in double quantum dots, thereby introducing fermionic reservoir engineering. We verify our prediction with a weak-coupling quantum master equation and the exact solution. Directionality is attained through the interference of coherent and dissipative coupling. The relative phase is tuned with an external magnetic eld, such that directionality can be reversed, as well as turned on and off dynamically.