The use of Ewald and related methods to handle electrostatic interactions in explicit-solvent simulations of solutions imposes an artificial periodicity on systems which are inherently nonperiodic. The consequences of this approximation should be assessed, since they may crucially affect the reliability of those computer simulations. In the present study, we propose a general method based on continuum electrostatics to investigate the nature and magnitude of periodicity-induced artifacts. As a first example, this scheme is applied to the solvation free-energy of a spherical ion. It is found that artificial periodicity reduces the magnitude of the ionic solvation free-energy, because the solvent in the periodic copies of the central unit cell is perturbed by the periodic copies of the ion, thus less available to solvate the central ion. In the limit of zero ionic radius and infinite solvent permittivity, this undersolvation can be corrected by adding the Wigner self-energy term to the solvation free-energy. For ions of a finite size or a solvent of finite permittivity, a further correction is needed. An analytical expression for this correction is derived using continuum electrostatics. As a second example, the effect of artificial periodicity on the potential of mean force for the interaction between two spherical ions is investigated. It is found that artificial periodicity results in an attractive force between ions of like charges, and a repulsive force between ions of opposite charges. The analysis of these two simple test cases reveals that two individually large terms, the periodicity-induced perturbations of the Coulomb and solvation contributions, often cancel each other significantly, resulting in an overall small perturbation. Three factors may prevent this cancellation to occur and enhance the magnitude of periodicity-induced artifacts: (i) a solvent of low dielectric permittivity, (ii) a solute cavity of non-negligible size compared to the unit cell size, and (iii) a solute bearing a large overall charge.