We report a new type of phononic crystals with topologically nontrivial band gaps for both longitudinal and transverse polarizations, resulting in protected one-way elastic edge waves. In our design, gyroscopic inertial effects are used to break the time-reversal symmetry and realize the phononic analogue of the electronic quantum (anomalous) Hall effect. We investigate the response of both hexagonal and square gyroscopic lattices and observe bulk Chern numbers of 1 and 2, indicating that these structures support single and multimode edge elastic waves immune to backscattering. These robust one-way phononic waveguides could potentially lead to the design of a novel class of surface wave devices that are widely used in electronics, telecommunication, and acoustic imaging. DOI: 10.1103/PhysRevLett.115.104302 PACS numbers: 46.40.Cd, 46.40.Ff, 73.43.-f Topological states in electronic materials, including the quantum Hall effect [1] and topological insulators [2,3], have inspired a number of recent developments in photonics [4,5], phononics [6][7][8][9], and mechanical metamaterials [10][11][12][13]. In particular, in analogy to the quantum anomalous Hall effect [14], one-way electromagnetic waveguides in two-dimensional systems have been realized by breaking time-reversal symmetry [15][16][17][18].Very recently, unidirectional edge channels have been proposed for elastic waves using Coriolis force in a noninertial reference frame [19], but such a rotating frame is very difficult to implement in solid state devices. Moreover, one-way propagation of scalar acoustic waves has also been proposed by introducing rotating fluids [20][21][22]. However, it is important to recognize that elastic waves in solids have both transverse and longitudinal polarizations, while acoustic waves in fluids are purely longitudinal. As a result, it is challenging to achieve topological protection for elastic waves on an integrated platform.Here, we present a robust strategy to create topologically nontrivial edge modes for both longitudinal and transverse polarizations in a solid medium. In particular, we introduce gyroscopic phononic crystals, where each lattice site is coupled with a spinning gyroscope that breaks timereversal symmetry in a well-controlled manner. In both hexagonal and square lattices, gyroscopic coupling opens band gaps that are characterized by Chern numbers of 1 and 2. As a result, at the edge of these lattices both single-mode and multimode one-way elastic waves are observed to propagate around arbitrary defects without backscattering.To start, we consider a hexagonal phononic crystal with equal masses (m 2 ¼ m 1 ) connected by linear springs [red and black rods in Figs. 1(a) and 1(b)]. The resulting unit cell has 4 degrees of freedom specified by the displacements of m 1 and m 2 (U ¼ ½u for wave vectors μ within the first Brillouin zone. Here, ω denotes the angular frequency of the propagating wave and M ¼ diagfm 1 ; m 1 ; m 2 ; m 2 g is the mass matrix. Moreover, K is the 4 × 4 stiffness matrix and is a functi...