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Over recent decades, many revolutionary research advances have proven that a deep understanding of the properties of materials can help us make innovations in mechanical, electronic, and optical fields. To modulate a tremendous range of properties, scientists have invented many artificial materials, which will be the basis of future technologies and devices. Analogously, a low-loss dielectric medium, called a photonic crystal (PtC), enables complete control over light propagation. Due to the existence of periodic potential, the electron movements in the crystal will be influenced by the Bragg scattering. Therefore, gaps may occur in the energy band structure of crystals, meaning that photonics are forbidden with certain energies in certain directions. If the material constants in the crystals are sufficiently different, we can obtain the complete band gaps, preventing light form propagating in any possible direction from any source. Since light can be prohibited in the band gap, it is possible to freely modulate the behavior of light in PtCs, e.g., designing the photonic metamaterials4–6 and low-loss optical devices. Following PtCs, the analogous concept of phononic crystals (PnCs) is proposed based on the concept that a structure with periodic elastic moduli and mass densities, resulting in elastic wave (acoustic wave) propagation, can be affected quite strongly. Many efforts have focused on designing wave dispersion through Bragg scattering or local resonances to explore important properties, such as the band gap, band edge states, and slow wave effects. Due to these fundamental properties, PnCs can be designed by selecting physical and geometrical parameters for different and surprising potential applications, e.g., damping, isolation, and rectification of acoustic or elastic waves. In addition, based on the design of equal-frequency contours, negative refraction, focusing, beam splitting, self-collimation, and acoustic diodes can be obtained. Furthermore, the introduction of defect modes (cavities and waveguides) leads to selective frequency filters, waveguides, wavelength demultiplexing devices, delayers, effective acoustic circuits, and so on. Moreover, designing structures with a deep subwavelength scale can dramatically change acoustic wave propagation, rendering it possible to achieve a superlens, high-resolution imaging, superabsorption, and a cloak via transformation acoustics. To achieve tunable and reversible PnCs, external physical stimuli such as electric or magnetic fields, tensile deformation, and variation of temperature or phase transformation play a fundamental role and provide more chances to better control the bulk waves, plate waves, and surface waves in the artificial periodic structures. Furthermore, PnCs represent an entirely novel aspect to develop new materials for manipulating heat transport by managing heat flow in the same manner as sound waves, rendering PnCs useful for future thermoelectric devices. In recent years, much effort has been aimed at simultaneous control of the acoustic and optical properties in the same system based on phoxonic crystals (PxCs). Such periodic materials, also known as phononic–photonic or optomechanical structures, have dual band gaps for both photons and phonons simultaneously, leading to quite promising applications, i.e., enhancing the acoustical–optical or optomechanical interactions, effectively manipulating photons with phonons,and colocalizing photonic and phononic resonances. Obviously, PxCs provide the interesting purpose of designing new compact acousto-optic and sensing devices, while retaining high-frequency phonons. Obviously, how to better control, localize, and guide the sound (acoustic waves) and light (electromagnetic waves) simultaneously is an appealing and challenging research topic. Throughout the developments of PtCs, PnCs, and PxCs, a great deal of interest has been devoted to material and structural parameter studies for modulating photons and phonons, based on different targets and physical properties. With the in-depth research for applications, however, we should answer the following two problems from the perspective of structural design. What structures have interesting photonic and phononic properties? How can we design a device with certain outstanding properties prepared from PtCs, PnCs, and PxCs?
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