This volume consists of invited lectures and seminars presented at the NATO Advanced Study Institute "The Infrared and Submillimetre Sky after COBE," which was held at the Centre de Physique Theorique of Les Houches (France) in March 1991. The school has been planned by a Scientific Organizing Committee. It was organized with the aim of providing students and young researchers with an up-to-date account of the Cosmic Microwave Background, the Cosmic Infrared Background (if any), and the infrared emission of the Galaxy, after the early results from COBE (Cosmic Background Explo rer). It was attended by about sixty researchers from many countries. The lectures and seminars represent a complete coverage of our present knowledge and understanding of: the Early Universe, Large-Scale Structure, Dust in Galaxies, Infrared to Submillimetre Backgrounds, CMB Anisotropies, complementary observations and instrumentation problems, etc. Most of these lectures are reproduced in this volume. Unfortunately, a few lecturers have chosen not to submit their manuscript. I would like to express my gratitude to the Scientific Affairs Division of NATO (North Atlantic Treaty Organization) and to the Theoretical Physics and Astrophysics Sections of the CNRS (Centre National de la Re cherche Scientifique) for their generous support. Further help was obtain ed from the DRET (Direction des Recherches, Etudes et Techniques), the CNES (Centre National d'Etudes Spatiales) and the IN2P3 (Institut Natio nal de Physique Nuc1eaire et de Physique des Particules), which I hereby gratefully acknowledge.