Spectra of highn and non-lown degrees

We survey known results on spectra of structures and on spectra of relations on computable structures. asking when the set of all highn degrees can be such a spectrum, and likewise for the set of nonlown degrees. We then repeat these questions specifically for linear orders and for relations on the computable dense linear order Q. New results include realizations of the set of nonlown Turing degrees as the spectrum of a relation on Q for all n ≥ 1, and a realization of the set of nonlown Turing degrees as the spectrum of a linear order whenever n ≥ 2. The state of current knowledge is summarized in a table in the concluding section.