This article provides a comprehensive review of the current state of research in cold and ultracold molecules, serving as an introduction to a Special Issue of the New Journal of Physics on the same topic. It highlights the potential of cold and ultracold molecules to revolutionize physical chemistry, few-body physics, and quantum simulations. The review covers fundamental science, including molecular structure, chemistry, and precision measurements, as well as technological advancements in experimental techniques for creating and manipulating these molecules. The article also discusses potential applications such as quantum computing, precise control of molecular dynamics, and nanolithography. The discussion is based on recent experimental and theoretical work, concluding with a summary of future directions and open questions in this rapidly expanding field.This article provides a comprehensive review of the current state of research in cold and ultracold molecules, serving as an introduction to a Special Issue of the New Journal of Physics on the same topic. It highlights the potential of cold and ultracold molecules to revolutionize physical chemistry, few-body physics, and quantum simulations. The review covers fundamental science, including molecular structure, chemistry, and precision measurements, as well as technological advancements in experimental techniques for creating and manipulating these molecules. The article also discusses potential applications such as quantum computing, precise control of molecular dynamics, and nanolithography. The discussion is based on recent experimental and theoretical work, concluding with a summary of future directions and open questions in this rapidly expanding field.