We present Laguerre Voronoi based subdivision algorithms for the quadrilateral and hexahedral meshing of particle systems within a bounded region in two and three dimensions, respectively. Particles are smooth functions over circular or spherical domains. The algorithm first breaks the bounded region containing the particles into Voronoi cells that are then subsequently decomposed into an initial quadrilateral or an initial hexahedral scaffold conforming to individual particles. The scaffolds are subsequently refined via applications of recursive subdivision (splitting and averaging rules). Our choice of averaging rules yield a particle conforming quadrilateral/hexahedral mesh, of good quality, along with being smooth and differentiable in the limit. Extensions of the basic scheme to dynamic re-meshing in the case of addition, deletion, and moving particles are also discussed. Motivating applications of the use of these static and dynamic meshes for particle systems include the mechanics of epoxy/glass composite materials, bio-molecular force field calculations, and gas hydrodynamics simulations in cosmology.