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A bulk of the research on piezoelectric systems in recent years can be classified into two categories, 1) studies of linear piezoelectric oscillator arrays, 2) studies of nonlinear piezoelectric oscillators. This dissertation derives novel linear and nonlinear modeling and estimation methods for such piezoelectric systems. In the first part, this work develops modeling and design methods for Piezoelectric Subordinate Oscillator Arrays (PSOAs) for the wideband vibration attenuation problem. PSOAs offer a straightforward and low mass ratio solution to cancel out the resonant peaks in a host structure's frequency domain. Further, they provide adaptability through shunt tuning, which gives them the ability to recover performance losses because of structural parameter errors. This dissertation studies the derivation of governing equations that result in a closed-form expression for the frequency response function. It also analyzes systematic approaches to assign distributions to the nondimensional parameters in the frequency response function to achieve the desired flat-band frequency response. Finally, the effectiveness of PSOAs under ideal and nonideal conditions are demonstrated in this dissertation through extensive numerical and experimental studies. The concept of performance recovery, introduced in empirical studies, gives a measure of the PSOA's effectiveness in the presence of disorder before and after capacitive tuning. The second part of this dissertation introduces novel modeling and estimation methods for nonlinear piezoelectric oscillators. Traditional modeling techniques require knowledge of the structure as well as the source of nonlinearity. Data-driven modeling techniques used extensively in recent times build approximations. An adaptive estimation method, that uses reproducing kernel Hilbert space (RKHS) embedding methods, can estimate the underlying nonlinear function that governs the system's dynamics. A model built by such a method can overcome some of the limitations of the modeling approaches mentioned above. This dissertation discusses (i) how to construct the RKHS based estimator for the piezoelectric oscillator problem, (ii) how to choose kernel centers for approximating the RKHS, and (iii) derives sufficient conditions for convergence of the function estimate to the actual function. In each of these discussions, numerical studies are used to show the RKHS based adaptive estimator's effectiveness for identifying linearities in piezoelectric oscillators. / Doctor of Philosophy / Piezoelectric materials are materials that generate an electric charge when mechanical stress is applied, and vice versa, in a lossless transformation. Engineers have used piezoelectric materials for a variety of applications, including vibration control and energy harvesting. This dissertation introduces (1) novel methods for vibration attenuation using an array of piezoelectric oscillators, and (2) methods to model and estimate the nonlinear behavior exhibited by piezoelectric materials at very high mechanical forces or electric charges. Arrays of piezoelectric oscillators attached to a host structure are termed piezoelectric subordinate oscillator arrays (PSOAs). With the careful design of PSOAs, we show that we can reduce the vibration of the host structure. This dissertation analyzes methodologies for designing PSOAs and illustrates their vibration attenuation capabilities numerically and experimentally. The numerical and experimental studies also illustrate the robustness of PSOAs. In the second part of this dissertation, we analyze reproducing kernel Hilbert space embedding methods for adaptive estimation of nonlinearities in piezoelectric systems. Kernel methods are extensively used in machine learning, and control theorists have studied adaptive estimation of functions in finite-dimensional spaces. In this work, we adapt kernel methods for adaptive estimation of functions in infinite-dimensional spaces that appear while modeling piezoelectric systems. We derive theorems that ensure convergence of function estimates to the actual function and develop algorithms for careful selection of the kernel basis functions.