We review different expressions that have been proposed for the stress tensor and for the linear momentum of light in dielectric media, focusing on the Abraham and Minkowski forms. Analyses of simple models and consideration of available experimental results support the interpretation of the Abraham momentum as the kinetic momentum of the field, while the Minkowski momentum is the recoil momentum of absorbing or emitting guest atoms in a host dielectric. Momentum conservation requires consideration not only of the momentum of the field and of recoiling guest atoms, but also of the momentum the field imparts to the medium. Different model assumptions with respect to electrostriction and the dipole force lead to different expressions for this momentum. We summarize recent work on the definition of the canonical momentum for the field in a dielectric medium.