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 $Q=1$ and a non-topological skyrmionium with $Q=0$ under the influence of an external field gradient. The $Q=1$ skyrmion undergoes Hall motion perpendicular to the direction of the field gradient with a drift velocity proportional to the gradient. In contrast, the non-topological $Q=0$ skyrmionium is accelerated in the direction of the field gradient, thus exhibiting ordinary Newtonian motion. When the external field is switched off the $Q=1$ skyrmion is spontaneously pinned around a fixed guiding center, whereas the $Q=0$ skyrmionium moves with constant velocity $v$. We give a systematic calculation of a skyrmionium traveling with any constant velocity $v$ that is smaller than a critical velocity $v_c$.
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