It is of both theoretical and engineering significance to understand the random packings of non-spherical particles. However, apart from the well-discussed aspect ratio, studies on particle shapes continuously evolved from sphere to ideal polyhedra are still lacking, which represents the roundness effect. In this work, we investigate two packing states, namely random close packing (RCP) and saturated random packing (SRP), of spheropolyhedra (SPP), including three shape families, namely spherotetrahedron (SPT), spherocube (SPC), and spherooctahedron (SPO). We observe a common density peak phenomenon of these two states for all the families with respect to sphericity. Specifically, the RCP densities can reach similar to 0.746, similar to 0.750, and similar to 0.731 for the SPT, SPC, and SPO respectively, comparable with the crystalline density similar to 0.74 for spheres. Density peaks of the SRP locate at the sphericity -0.96 for all the families. Additionally, the local structural analysis reveals the complex dependencies of order parameters on the roundness, including positional order q(6) and facet alignment Delta. The SRP states are more random in particle position than orientation. The dimer clusters formed by particles sharing common facets are also explored. We find that the facet number of a single particle is positively correlated with the q(6) yet negatively correlated with the Delta and the cluster ratio for all the polyhedron-like shapes at the RCP. Furthermore, the mechanism of excluded volume can explain the density peak of both the RCP and SRP for all the families and even partly reproduce the general trend of the RCP density for SPT. (C) 2019 Elsevier B.V. All rights reserved.