We define the notion of a cardinal spectrum, as a class-valued cardinal function. After a discussion of various contexts in which such functions are studied, we take up the example of the universality spectrum and explain its relevance to certain problems in set theory and model theory. Through this example, we explain what sort of problems one may face when trying to calculate such spectra, mostly having to do with difficulties in providing independence results in set theory at certain sort of cardinals (successors of singulars). We then describe some results (including a joint result by S. Shelah and the speaker) which resolve some of the issues arising in this context, and give directions of present research (joint with C. Morgan).