The research of this dissertation is formulated in two fields, i.e., the theoretical and computational studies of circumferential wrinkling on soft nanofibers and the swelling mechanics study of a bi-layered spherical hydrogel containing a hard core. Continuous polymer nanofibers have been massively produced by means of the low-cost, top-down electrospinning technique. As a unique surface instability phenomenon, surface wrinkling in circumferential direction is commonly observed on soft nanofibers in electrospinning. In this study, a theoretical continuum mechanics model is developed to explore the mechanisms of circumferential wrinkling on soft nanofibers under uniaxial stretching. The model is able to examine the effects of elastic properties, surface energy, and fiber radius on the critical axial stretch to trigger circumferential wrinkling and to discover the threshold fiber radius to initiate spontaneous wrinkling. In addition, nonlinear finite element method (FEM) is further adopted to predict the critical mismatch strain to evoke circumferential wrinkling in core-shell polymer nanofibers containing a hard core, as a powerful computational tool to simulate controllable wrinkling on soft nanofibers via co-electrospinning polymer nanofibers incorporated with nanoparticles as the core. The studies provide rational understanding of surface wrinkling in polymer nanofibers and technical approaches to actively tune surface morphologies of polymer nanofibers for particular applications, e.g. high-grade filtration, oil-water separation, polymer nanocomposites, wound dressing, tissue scaffolding, drug delivery, and renewable energy harvesting, conversion, and storage, etc. Furthermore, hydrogels are made of cross-linked polymer chains that can swell significantly when imbibing water and exhibit inhomogeneous deformation, stress, and, water concentration fields when the swelling is constrained. In this study, a continuum mechanics field theory is adopted to study the swelling behavior of a bi-layered spherical hydrogel containing a hard core. The problem is reduced into a two-point boundary value problem of a 2nd-order nonlinear ordinary differential equation (ODE) and solved numerically. Effects of material properties on the deformation, stress, and water concentration fields of the hydrogel are examined. The study offers a rational route to design and regulate hydrogels with tailorable swelling behavior for practical applications in drug delivery, leakage blocking, etc.