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We study the dynamics of skyrmions in Dzyaloshinskii-Moriya materials with easy-axis anisotropy. An important link between topology and dynamics is established through the construction of unambiguous conservation laws obtained earlier in connection with magnetic bubbles and vortices. In particular, we study the motion of a topological skyrmion with skyrmion number and a nontopological skyrmionium with under the influence of an applied field gradient. The skyrmion undergoes Hall motion perpendicular to the direction of the field gradient with a drift velocity proportional to the gradient. In contrast, the nontopological skyrmionium is accelerated in the direction of the field gradient, thus exhibiting ordinary Newtonian motion. When the applied field is switched off the skyrmion is spontaneously pinned around a fixed guiding center, whereas the skyrmionium moves with constant velocity  …