I. INTRODUCTIONThe past few decades have witnessed remarkable developments of laser techniques setting the stage for new areas of research in molecular physics. It is now possible to interrogate molecules in the ultrafast and ultracold regimes of molecular dynamics and the measurements of molecular structure and dynamics can be made with unprecedented precision. Molecules are inherently complex quantum-mechanical systems. The complexity of molecular structure, if harnessed, can be exploited for yet another step forward in science, potentially leading to technology for quantum computing, quantum simulation, precise field sensors, and new lasers.The goal of the present article is to review the major developments that have led to the current understanding of molecule -field interactions and experimental methods for manipulating molecules with electromagnetic fields. Molecule -field interactions are at the core of several, seemingly distinct, areas of molecular physics. This is reflected in the organization of this article, which includes sections on Field control of molecular beams, External field traps for cold molecules, Control of molecular orientation and molecular alignment, Manipulation of molecules by nonconservative forces, Ultracold molecules and ultracold chemistry, Controlled many-body phenomena, Entanglement of molecules and dipole arrays, and Stability of molecular systems in high-frequency super-intense laser fields. By combining these topics in the same review, we would like to emphasize that all this work is based on the same basic Hamiltonian.This review is also intended to serve as an introduction to the excellent collection of articles appearing in this same-titled volume of Molecular Physics [1-27]. These original contributions demonstrate the latest developments exploiting control of molecules with electromagnetic fields. The reader will be treated to a colourful selection of articles on topics as diverse as Chemistry in laser fields, Quantum dynamics in helium droplets, Effects of microwave and laser fields on molecular motion, Rydberg molecules, Molecular structure in external fields, Quantum simulation with ultracold molecules, and Controlled molecular interactions, written by many of the leading protagonists of these fields.This article is concerned chiefly with the effects of electromagnetic fields on low-energy rotational, fine-structure and translational degrees of freedom. There are several important research areas that are left outside the scope of this paper, most notably the large body of work on the interaction of molecules with attosecond laser pulses and high harmonic generation [28], coherent control of molecular dynamics [29] and optimal control of molecular processes [30]. We limit the discussion of resonant interaction of light with molecules to laser cooling strategies. We do not survey spectroscopy or transfer of population between molecular states. Even with these restrictions, this is a vast area to review, as is apparent from the number of references. We did our best to in...