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The present notes are organized as the lectures given at the Les Houches Summer School "Quantum Optics and Nanophotonics" in August 2013. The first section contains an introduction and a description of the current state-of-the-art for Casimir force measurements and their comparison with theory. The second and third sections are a pedagogical presentation of the main features of the theory of Casimir forces for 1-dimensional model systems and for mirrors in 3-dimensional space.

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