Complex wrinkle patterns have been observed in various thin film systems, typically with integrated hard and soft materials for various technological applications as well as in nature. The underlying mechanism of wrinkling has been generally understood as a stress-driven instability. On an elastic substrate, equilibrium and energetics set the critical condition and select the wrinkle wavelength and amplitude. On a viscous substrate, wrinkles grow over time and kinetics select the dominant wavelength. More generally, on a viscoelastic substrate, both energetics and kinetics play important roles in determining the critical condition, the growth rate, and wrinkle patterns. The dynamics of wrinkling, while analogous to other phase ordering phenomena, is rich and distinct under the effects of a variety of stress conditions and nonlocal film-substrate interactions. In this study, a new mathematical model is developed for wrinkling of isotropic and anisotropic elastic films on viscoelastic substrates. Analytic solutions are obtained by a linear perturbation analysis and a nonlinear energy minimization method, which predict the kinetics of wrinkle growth at the initial stage and the equilibrium states at the long-time limit, respectively. In between, a power-law coarsening of the wrinkle wavelength is predicted by a scaling analysis. Numerical simulations confirm the analytical predictions and show diverse wrinkle patterns under various stress conditions. For isotropic elastic films, a transition from parallel wrinkles to zigzag patterns is predicted under anisotropic biaxial stresses. For cubic crystal films, the anisotropic elastic property leads to formation of orthogonal wrinkle patterns under equi-biaxial stresses. In general, the competition between the stress anisotropy and the material anisotropy controls the evolution of wrinkle patterns. Based on the mathematical model, two potential applications of the wrinkling phenomenon are explored, one for surface patterning and the other for estimating viscoelastic properties of thin polymer films. The theoretical and numerical results from this study are compared with experimental observations that are available in literature and through collaborations with experimental groups. The last chapter of this dissertation considers ratcheting-induced wrinkling for an elastic film on an elastoplastic substrate under cyclic temperatures, demonstrating an analogy between plastic ratcheting and viscous creep. / text

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