This article reviews the current status of lattice-dynamical calculations in crystals using density-functional perturbation theory (DFPT), with a focus on the plane-wave pseudo-potential method. It covers several specialized topics, including the implementation for metals, the calculation of responses to macroscopic electric fields and their relevance to long wavelength vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodology is demonstrated through a number of applications in the literature. The article begins with an introduction to lattice dynamics from electronic-structure theory, followed by a detailed explanation of density-functional theory (DFT) and its key components, such as the Kohn-Sham equations and the local-density approximation. It then discusses linear response theory, monochromatic perturbations, homogeneous electric fields, and the relation to the variational principle. The response to strain deformations and higher-order responses are also covered, including the 2n+1 theorem and nonlinear susceptibilities. The implementation of DFPT within the plane-wave pseudo-potential scheme is described in detail, including the use of ultra-soft pseudo-potentials and localized basis sets. Other approaches, such as the dielectric approach, frozen phonons, and vibrational properties from molecular dynamics, are also discussed. The article provides a comprehensive overview of significant applications of DFPT to the physics of insulators and metals, including their surfaces, alloys, and microstructures. It covers various materials, such as simple semiconductors, simple metals, oxides, and other materials, and discusses topics like phonons in bulk crystals, phonons in semiconductor alloys and superlattices, lattice vibrations at surfaces, soft phonons, and pressure-induced lattice transformations. Finally, the article concludes with a discussion of thermal properties of crystals and surfaces, anharmonic effects, isotopic broadening of Raman lines, and vibrational broadening of electronic core levels.This article reviews the current status of lattice-dynamical calculations in crystals using density-functional perturbation theory (DFPT), with a focus on the plane-wave pseudo-potential method. It covers several specialized topics, including the implementation for metals, the calculation of responses to macroscopic electric fields and their relevance to long wavelength vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodology is demonstrated through a number of applications in the literature. The article begins with an introduction to lattice dynamics from electronic-structure theory, followed by a detailed explanation of density-functional theory (DFT) and its key components, such as the Kohn-Sham equations and the local-density approximation. It then discusses linear response theory, monochromatic perturbations, homogeneous electric fields, and the relation to the variational principle. The response to strain deformations and higher-order responses are also covered, including the 2n+1 theorem and nonlinear susceptibilities. The implementation of DFPT within the plane-wave pseudo-potential scheme is described in detail, including the use of ultra-soft pseudo-potentials and localized basis sets. Other approaches, such as the dielectric approach, frozen phonons, and vibrational properties from molecular dynamics, are also discussed. The article provides a comprehensive overview of significant applications of DFPT to the physics of insulators and metals, including their surfaces, alloys, and microstructures. It covers various materials, such as simple semiconductors, simple metals, oxides, and other materials, and discusses topics like phonons in bulk crystals, phonons in semiconductor alloys and superlattices, lattice vibrations at surfaces, soft phonons, and pressure-induced lattice transformations. Finally, the article concludes with a discussion of thermal properties of crystals and surfaces, anharmonic effects, isotopic broadening of Raman lines, and vibrational broadening of electronic core levels.