Zeno's paradoxes are among the oldest puzzles at the intersection of mathematics, philosophy, and physics (in alphabetical order). The traditional resolution of Zeno's paradoxes of motion involves modeling them in terms of the real line and interpreting the iterated procedure as an infinite series.
As pointed out in one of the comments, Heisenberg's uncertainty principle provides another way of accounting for the puzzle, by arguing that it has no physical meaning.
H. Jerome Keisler in his article "The hyperreal line" (207–237) in the collection
Real numbers, generalizations of the reals, and theories of continua. Edited by Philip Ehrlich. Synthese Library, 242. Kluwer Academic Publishers Group, Dordrecht, 1994
provides a different mathematical resolution of the puzzle in terms of the hyperreal continuum.
More recently (2013), Terry Tao notes the mathematical significance of these paradoxes by noting that they "make the important point that real analysis cannot be reduced to a branch of discrete mathematics, but requires additional tools in order to deal with the continuum" (see http://www.ams.org/mathscinet-getitem?mr=3026767).
In a review of Graham Oppy's book, John H. Mason makes the following intriguing comment, indicative of the richness of the issues involved: Have you ever briefly called upon Zeno's paradoxes when introducing the notion of limit to students? For example, the fact that Achilles really does catch the tortoise is only because he crosses distances halving in length in intervals of time also halving in length; the arrow does actually get to its target, even though it has to surmount an infinite number of decreasingly small intervals. This book addresses these and many other paradoxes involving infinitely large and infinitely small quantities with philosophical precision and reasoning. It reveals that there are much larger issues at stake than are perhaps commonly recognised, and certainly than are `dismissed' with the Cauchy-Weierstrass formalism of limits. See http://www.ams.org/mathscinet-getitem?mr=2238333