5 In this paper, we generate both packings of binary superdisks in R 2 and monodisperse superballs in R 3 that represent the maximally random jammed (MRJ) state of these parti- cles, using a novel event-driven molecular dynamics algorithm [34, 35] and investigate their Optimal packings of congruent superdisks and superballs apparently are realized by cer- tain Bravais lattices possessing symmetries consistent with those of the particles [25, 26, 28]. Even these crystalline packings exhibit rich characteristics that are distinctly different from other known packings of nonspherical particles. For example, we found that the maximal density φ$_{max}$ as a function of p at p = 1 (the sphere or circular-disk point) is nonanalytic and increases dramatically as p moves away from unity. In addition, we have discovered two-fold degenerate maximal density states for square-like superdisks, and both cube-like and octahedron-like superballs. the two-dimensional ( d = 2) and three-dimensional ( d = 3) cases, respectively. A superdisk possesses square symmetry, as p moves away from unity, two families of superdisks can be obtained, with the symmetry axes rotated 45 degrees with respect to each other; when p < 0 . 5, the superdisk is concave (see Fig. 1). A superball can possess two types of shape anisotropy: cube-like shapes (for p > 1) and octahedron-like shapes (for 0 < p < 1) with a shape change from convexity to concavity as p passes downward through 0 . 5 (see Fig. 2). FIG. 2: (color online). Superballs with different values of the deformation parameter p . FIG. 1: (color online). Superdisks with different values of the deformation parameter p .