This thesis presents the study of the topological properties in the atomic vibrations in solids. The state-of-the-art quantum mechanical simulation techniques are presented first to accurately describe the band structures of electrons and vibrational spectra of nuclei for topological analysis. Two concepts are introduced to investigate the non-trivial band crossing in the vibrational spectra. Depending the number of phonon bands and the intrinsic crystalline symmetry, the topological invariant carried by the band crossing can either be an integer number for a two-band subspace or an non-Abelian frame charge when at least three bands are involved. The phonon band crossing formed in a two-band subspace is simply a replica of topological semimetals in electronic systems, although the intrinsic properties of phonons, including the preservation of time-reversal symmetry and the accessibility of the bosonic excitation spectra, indicate that such single-gap topological properties are ubiquitous in phonons. Taking a step further, when three or more bands are included, these intrinsic properties provide unique advantages for phonons to fulfill the requirements for multi-gap topology, rendering phonons as the primary platform to study non-Abelian braiding. The possible experimental signatures are also described and predicted.