Generalized harmonic analysis represents the culmination and combination of a number of very diverse mathematical movements. The spectrum theory of the present paper has as one very special application the theory of almost periodic functions. It is not difficult to prove that the spectrum of such a function contains a discrete set of lines and no continuous part, and to deduce from this, Bohr's form of the Parseval theorem. The transition from the Parseval theorem to the Weierstrassian theorem that it is possible to approximate uniformly to any almost periodic function by a sequence of trigonometrical polynomials follows essentially laid down by Weyl, though it differs somewhat in detail. Besides the well-known generalizations of almost periodic functions due to Stepanoff, Besicovitch, Weyl, and the author, there is the little explored field of extensions of almost periodic functions containing a parameter. These have been used by Mr. C. F. Muckenhoupt to prove the closure of the set of the Eigenfunktionen of certain linear vibrating systems. This is one of the few applications of almost periodic functions of a fairly general type to definite mathematicophysical problems. Our last section is devoted to this, and to related matters.