This dissertation studies the thermodynamic and structural properties of aqueous dispersions of ionic microgels – soft colloidal particles composed of cross-linked polymer gels that swell in a good solvent. Starting from a coarse-grained model of microgel particles, we perform computer simulations and theoretical calculations using two complementary implementations of Poisson- Boltzmann (PB) theory. Within the framework of a cell model, the nonlinear PB equation is exactly solved and used to compute counterion distributions and osmotic pressures. By varying the free energy with respect to microgel size, we obtain exact statistical mechanical relations for the electrostatic component of the single-particle osmotic pressure. Explicit results are presented for equilibrium swelling ratios of charged microcapsules and of charged cylindrical and spherical microgels with fixed charge uniformly distributed over the surface or volume of the particle. Molecular dynamics simulations validate the theoretical findings. In the second method, within a one-component model framework, based on a linear-response approximation for effective electro- static interactions, we develop Monte Carlo (MC) simulations to compute the equilibrium swelling ratio, bulk osmotic pressure, radial distribution function, and static structure factor. Results presented in this dissertation demonstrate that swelling of ionic microgels increases with increasing microgel charge and decreases with increasing concentration of salt and microgels. In addition, results demonstrate that the microion distributions and osmotic pressure determine equilibrium swelling of microgels. Cell model predictions for bulk osmotic pressure agree well with data from MC simulations of the one-component model. The MC simulations also provide access to structural properties and to swelling behavior of microgels in highly concentrated suspensions. Taken together, results obtained in this work provide insight into factors of importance for practical use of microgels as drug delivery systems, in tissue engineering, and for other biomedical applications.