The Born model of ionic solvation assumes that the solvent is a continuum which has a linear response and does not explicitly include nonlinear effects such as dielectric saturation and electrostriction. Nonetheless, the Born model has been surprisingly successful in estimating the solvation energies of ions in solution. Recently, we developed a simple quasicontinuum theory of ionic solvation, referred to here as the HBI model [J.-K. Hyun, C. S. Babu, and T. Ichiye, J. Phys. Chem. 99, 5187 (1995)]. The HBI model exhibits dielectric saturation, which is demonstrated in the expressions for the average orientation of the solvent molecules and the solvation energy, but does not incorporate the effects of electrostriction. In this article, new models of ionic solvation are developed that exhibit not only dielectric saturation but also electrostriction. Expressions for the radial and orientational distribution functions of the solvent molecules, and the solvation energy are derived and compared with those from the HBI and Born models to examine the characteristics of different models of ionic solvation. Comparisons are also made to molecular dynamics simulations of ions in aqueous solution, where the complex molecular effects make separation of different nonlinear effects difficult. In particular, the systematic discrepancies in the solvation energies from the different models are used to understand how different nonlinear effects contribute to the response when water is the solvent. For small, monovalent ions, the success of the Born model is attributed not to the lack of dielectric saturation or electrostriction, but rather to the cancellation of one effect by the other.