Part I Foundations of Casimir Theory.- Casimir's Calculation.- Macroscopic Quantum Electrodynamics.- The Quantum Nature of the Casimir Force.- Part II Surprises in Casimir Theory.- The Cut-off-dependence of the Casimir Force.- The Divergence of the Casimir Stress.- Part III Conundrums in Casimir Theory.- The Casimir Force in a Compressive' Medium.- The Casimir Force in Maxwell's Fish-Eye.- Outlook.- Appendices.