Hilbert’s lectures on radiation theory (Hilbert 1912, this Volume, Chapter 5) contain a discussion of Planck’s radiation formula and also comment on the underlying quantum hypothesis. But it is in two lecture courses that occur 10 and 14 years later that Hilbert explicitly concerns himself with the old quantum theory and the new quantum mechanics. A decade passed before, the first lecture course was given under the title “Mathematical Foundations of Quantum Theory” (Hilbert 1922/23a) in the winter semester 1922/23 and was worked out by L. Nordheim and G. Heckmann. It presents what is now known as the “old quantum theory,” especially Bohr’s quantization rules using the framework of Hamilton-Jacobi theory and action-angle variables in classical mechanics.