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 discusses various specialized topics, including the implementation for metals, the calculation of the response 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 various applications in the literature. The article begins with an introduction to lattice vibrations and their importance in solid-state physics. It then outlines the theoretical framework of DFPT, including lattice dynamics from electronic-structure theory, density-functional theory, and linear response. The paper discusses the relationship between electronic and lattice-dynamical properties, emphasizing the importance of these connections in calculating lattice-dynamical properties of specific systems. The article then presents the implementation of DFPT, including the use of plane waves and pseudo-potentials, ultra-soft pseudo-potentials, and localized basis sets. It also covers other approaches such as the dielectric approach, frozen phonons, and vibrational properties from molecular dynamics. The applications section discusses phonons in bulk crystals, semiconductor alloys and super-lattices, lattice vibrations at surfaces, soft phonons and pressure-induced lattice transformations, thermal properties of crystals and surfaces, anharmonic effects, isotopic broadening of Raman lines, and vibrational broadening of electronic core levels. The paper concludes with a discussion of the current state of theoretical condensed-matter physics and computational materials science, highlighting the ability to calculate specific properties of specific materials using ab initio quantum mechanical techniques. It emphasizes the importance of DFPT in understanding phonons and their role in various physical properties of solids. The article also discusses the challenges and limitations of DFPT, including the need for accurate approximations of the exchange-correlation energy and the difficulties in handling strongly correlated systems. Overall, the paper provides a comprehensive overview of DFPT and its applications in the study of lattice vibrations and related properties in solids.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 discusses various specialized topics, including the implementation for metals, the calculation of the response 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 various applications in the literature. The article begins with an introduction to lattice vibrations and their importance in solid-state physics. It then outlines the theoretical framework of DFPT, including lattice dynamics from electronic-structure theory, density-functional theory, and linear response. The paper discusses the relationship between electronic and lattice-dynamical properties, emphasizing the importance of these connections in calculating lattice-dynamical properties of specific systems. The article then presents the implementation of DFPT, including the use of plane waves and pseudo-potentials, ultra-soft pseudo-potentials, and localized basis sets. It also covers other approaches such as the dielectric approach, frozen phonons, and vibrational properties from molecular dynamics. The applications section discusses phonons in bulk crystals, semiconductor alloys and super-lattices, lattice vibrations at surfaces, soft phonons and pressure-induced lattice transformations, thermal properties of crystals and surfaces, anharmonic effects, isotopic broadening of Raman lines, and vibrational broadening of electronic core levels. The paper concludes with a discussion of the current state of theoretical condensed-matter physics and computational materials science, highlighting the ability to calculate specific properties of specific materials using ab initio quantum mechanical techniques. It emphasizes the importance of DFPT in understanding phonons and their role in various physical properties of solids. The article also discusses the challenges and limitations of DFPT, including the need for accurate approximations of the exchange-correlation energy and the difficulties in handling strongly correlated systems. Overall, the paper provides a comprehensive overview of DFPT and its applications in the study of lattice vibrations and related properties in solids.