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This tutorial chapter provides an introduction to the vibrant field of cold molecular collisions in the presence of external electromagnetic fields. The kinetic energy of molecules confined in electromagnetic traps is small compared to their interactions with external fields, so the influence of molecule -field interactions on molecular collisions at ultralow temperatures can no longer be neglected or treated perturbatively. The intricate topology of spin-dependent, multidimensional intermolecular interactions makes it necessary to use accurate, non-perturbative quantum scattering techniques for solving the Schrödinger equation for molecules colliding in the presence of external fields. The primary goal of this chapter is to provide an overview of multichannel quantum scattering theory of molecular collisions in external fields, with a focus on rigorous coupled-channel methodology applied to molecules of current experimental interest. We also present an overview of the mechanisms for controlling elastic and inelastic molecular collisions with external magnetic fields.

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