In this work we present parallelotope bundles, i.e., sets of parallelotopes for a symbolic representation of polytopes. We define a compact representation of these objects and show that any polytope can be canonically expressed by a bundle. We propose efficient algorithms for the manipulation of bundles. Among these, we define techniques for computing tight over-Approximations of polynomial transformations. We apply our framework, in combination with the Bernstein technique, to the reachability problem for polynomial dynamical systems. The accuracy and scalability of our approach are validated on a number of case studies. © 2016 ACM.