  In 1952, Heinrich Scholz published a question in the Journal of Symbolic
Logic asking for a characterization of spectra, i.e., sets of natural numbers
that are the cardinalities of finite models of first order sentences. G\"unter
Asser asked whether the complement of a spectrum is always a spectrum. These
innocent questions turned out to be seminal for the development of finite model
theory and descriptive complexity. In this paper we survey developments over
the last 50-odd years pertaining to the spectrum problem. Our presentation
follows conceptual developments rather than the chronological order. Originally
a number theoretic problem, it has been approached in terms of recursion
theory, resource bounded complexity theory, classification by complexity of the
defining sentences, and finally in terms of structural graph theory. Although
Scholz' question was answered in various ways, Asser's question remains open.
One appendix paraphrases the contents of several early and not easily accesible
papers by G. Asser, A. Mostowski, J. Bennett and S. Mo. Another appendix
contains a compendium of questions and conjectures which remain open.
