<p>Given the vast differences in interaction details, describing the dynamics of structurally disordered materials in a unified theoretical framework presents a fundamental challenge to condensed-matter physics and materials science. Here we numerically investigate a double-percolation scenario for the two most important relaxation processes of supercooled liquids and glasses, the so-called α and β relaxations. For several simple glass formers, we find that when monitoring the dynamic shear modulus as temperature is lowered from the liquid state, percolation of immobile particles takes place at the temperature locating the α process. Mirroring this, upon continued cooling into the glass state, the mobile-particle percolation transition pinpoints a β process whenever the latter is well separated from the main (α) process. For two-dimensional systems under the same conditions, percolation of mobile and immobile particles occurs nearly simultaneously, and no β relaxation can be identified. Our findings suggest that a general description of glassy dynamics should be based on a percolation perspective.</p>