Exact configurations of the four-dimensional Skyrme model are presented. The static configurations have the profile. Which behaves as a kink and, consequently, the corresponding energy-momentum tensor describes a domain wall. Furthermore, a class of exact time-periodic Skyrmions is discovered. Within such a class, it is possible to disclose a remarkable phenomenon that is a genuine effect of the Skyrme term. For a special value of the frequency the Skyrmions admit a nonlinear superposition principle. One can combine two or more exact "elementary" Skyrmions (which may depend in a nontrivial way on all the spacelike coordinates) into a new exact composite Skyrmion. Because of such a superposition law, despite the explicit presence of nonlinear effects in the energy-momentum tensor, the interaction energy between the elementary Skyrmions can be computed exactly. The relations with the appearance of Skyrme crystals are discussed.