Statistical theories of animal movement have often been based on models of random walks where movements take place in discrete steps and occur at discrete times.  The multiplicity of distributions required in these approaches to describe animal movement (i.e., distributions of: angles, discrete steps, and times), have effects that cannot be simply disentangled, and hence cannot be unambiguously determined. I present a mathematical formulation of continuous animal movements. A single time dependent distance statistic, the mean square displacement, which may be directly measured or modelled, is a central determinant of such random walks and encapsulates key information about the statistical properties of animal movements. The model allows prediction of what were previously independent determinants of the motion. Because of the new emphasis on the mean square displacement, this work may open up a new set of questions concerning animal movements and related phenomena. The results are dir