Displacement measurements are found everywhere, both in scientific applications and in the everyday life. Classical physics and experience suggest that one can perform these measurements with unlimited precision upon technological improvements, without perturbing the measured system. Quantum physics, instead, changes this picture, predicting that the more precise a displacement measurement is done, the larger the disturbance, or quantum backaction, affecting the momentum of the measured system. As of today, the most precise displacement measurements are done by reflecting a laser field off a mechanical resonator and interferometrically measuring the phase of that field. This interaction, stemming from radiation pressure forces, is at the heart of the field of optomechanics. In such a displacement measurement, the imprecision and the quantum backaction arise from the quantum fluctuations of the optical phase and amplitude quadratures, respectively. In addition, mechanical systems unavoidably couple to a thermal environment, which introduces more disturbance and hinders the observation of quantum effects of the measurement. Nevertheless, an efficient quantum measurement can be realized whenever the information about the displacement is gathered at a rate close to the one at which the mechanical resonator is perturbed, due to both thermal forces and the quantum backaction. When available, the result of this measure-ment can be used to purify the state-of-knowledge held by an observer about the mechanics, that is, the conditional state. Based on this knowledge, the observer can exert a measurement-based quantum control to convert this conditional state into an unconditional one. In this thesis, we report experiments achieving quantum displacement measurements of a soft-clamped membrane resonator, inserted in the middle of an optical cavity. The cornerstones of the experiments are the extremely low dissipation rate of the mechanical energy and the high total detection efficiency, which together result in ameasurement efficiency of up to 56%. This corresponds to a system operating at the Heisenberg measurement-disturbance limit to within 33%. Furthermore, we employ the quantum trajectory formalism and a retrodiction measurement to experimentally verify the conditionalstate, which is a coherent one with purity of 78%. Based on the measurement outcomes, we design a feedback loop to exert a viscous force on the resonator. This feedback cools the mechanical mode down to its ground state, with a residual occupation of 0.29 phonons, thus realizing a long-standing goal in the field. Quantum measurements form an important tool for several applications, from ultra-precise sensing to the generation of entangled states. We exploit these quantum-limited measurements to perform displacement sensing below the standard quantum limit and to generate and verify the entanglement between two lasers, stemming from the simultaneous measurement of a common mechanical motion. The results shown in this thesis make this optomechanical platform attractive for further applications, such as the quantum transduction of information via an electro-opto-mechanical system and the generation of non-classical mechanical states by measurements.