A hydrogel consists of a cross-linked polymer network and solvent molecules, capable of large, reversible deformation in response to a variety of external stimuli. In particular, diverse instability patterns have been observed experimentally in swelling hydrogels under mechanical constraints. The present study develops a general theoretical framework based on a variational approach, which leads to a set of governing equations coupling mechanical and chemical equilibrium conditions for swelling deformation of hydrogels, along with proper boundary conditions. A specific material model is employed for analytical and numerical studies, for which the nonlinear constitutive behavior of the hydrogel is derived from a free energy function combining rubber elasticity with a polymer solution theory. A finite element method is then developed and implemented as a user-defined material (UMAT) in the commercial package, ABAQUS. By numerical simulations, the effect of constraint on inhomogeneous swelling of substrate-attached hydrogel lines is elucidated. It is found that crease-like surface instability occurs when the width-to-height aspect ratio of the hydrogel line exceeds a critical value. Next, by considering a hydrogel layer on a rigid substrate, swell-induced surface instability is studied in details. A linear perturbation analysis is performed to predict the critical condition for onset of the surface instability. In contrast to previously suggested critical conditions, the present study predicts a range of critical swelling ratios, from about 2.5 to 3.4, depending on the material properties of the hydrogel system. A stability diagram is constructed with two distinct regions for stable and unstable hydrogels with respect to two dimensionless material parameters. Numerical simulations are presented to show the swelling process, with evolution of initial surface perturbations followed by formation of crease-like surface patterns. Furthermore, with combined swelling and mechanical compression, the stability analysis is extended to predict a general critical condition that unifies the swell-induced surface instability of hydrogels with mechanically induced surface instability of rubbers. The effect of surface tension is found to be critical in suppressing short-wavelength modes of surface instability, while the substrate confinement suppresses long-wavelength modes. With both surface tension and substrate confinement, an intermediate wavelength is selected at a critical swelling ratio for onset of surface instability. Both the critical swelling ratio and the characteristic wavelength depend on the initial thickness of the hydrogel layer as well as other material properties of the hydrogel. It is found that the hydrogel layer becomes increasingly stable as the initial layer thickness decreases. A critical thickness is predicted, below which the hydrogel layer swells homogeneously and remains stable at the equilibrium state. Finally, three-dimensional finite element models are developed to simulate swelling deformation of hydrogel lines. Depending on the aspect ratio of the cross section as well as the material properties of the hydrogel, two types of swell-induced instability patterns are envisaged, i.e., localized surface instability versus global buckling. / text