Chemical shifts of Cs$^+$ nuclear resonances have been measured in aqueous salt solutions and in a wide variety of non-aqueous, mixed, and deuterated solvents. Shifts are found to be nonlinear with concentration down to 0.01 mol/dm$^3$, the precise degree of nonlinearity depending upon the dielectric constant of the solvent. A priori calculations of shift magnitudes, based on Ramsay's shielding formula and a postulated short-range radial distribution function, have yielded calculated shifts in reasonable agreement with observed values. The physical interaction responsible for variations of shift with concentration appears to be repulsive overlap of ion wave functions. The nonlinear functional relation between shift and molar concentration results from the influence of the ionic atmosphere on the probability of interionic contact. The form of this nonlinearity is accurately predicted by the postulated radial distribution function.