Dense packings have served as useful models of the structure of liquid, glassy and crystal states of matter, granular media, heterogeneous materials, and biological systems. Probing the symmetries and other mathematical properties of the densest packings is a problem of long-standing interest in discrete geometry and number theory. The preponderance of previous work has focused on spherical particles, and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the Adaptive Shrinking Cell (ASC) scheme. This novel optimization problem is solved here (using a variety of multi-particle initial configurations) to find dense packings of each of the Platonic solids in three-dimensional Euclidean space. We find the densest known packings of tetrahedra, octahedra, dodecahedra and icosahedra with densities $0.782...$, $0.947...$, $0.904...$, and $0.836...$, respectively. Unlike the densest tetrahedral packing, which must be a non-Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. Our simulations results, rigorous upper bounds that we derive, and theoretical arguments lead us to the strong conjecture that the densest packings of the Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This is the analog of Kepler's sphere conjecture for these solids.