A packing is a large collection of non-overlapping geometrical objects (e.g., hard spheres). Packing problems, such as how densely a given shape can fill 3D space, are usually easy to pose but notoriously difficult to solve mathematically. Here, we make an attempt to solve packing problems using a physical approach, by leveraging the connection between dense packings of hard particles and condensed phases (e.g., glassy and crystalline) of matter. This talk focuses on the quest for the general organizational principles of maximally dense (MD) and maximally random jammed (MRJ) packings of anisotropic (non-spherical) hard particles, which are respectively the crystalline and glassy states of the corresponding hard-particle colloids. Under equilibrium conditions, hard particles can self-organize into distinct ordered maximally dense packings via entropy-driven phase transitions. Kepler-type conjectures connecting the geometrical and symmetry characteristics of the particles and the self-organized ordered packing structures will be devised, which have ramifications in the design and fabrication of novel materials. The maximally random jammed packings are special non-equilibrium states of hard-particle colloids that possess hyperuniform quasi-long-range correlations only manifested on large length scales. Such a “hidden” order, also identified in other disordered systems including the density fluctuations in early universe, non-interacting fermion systems, many soft colloids, amorphous 2D materials, and even chicken retina photoreceptor patterns, can be considered as a universal signature of a new state of matter and has led to the development of disordered photonic materials with large isotropic band gaps.