Abstract: The investigation of computability in topological structures develops in some aspects similar to the investigation of computability in algebraic structures. If a countable algebraic structure is not computably presentable then its ‘‘degree of non-computability’’ is measured by the so called degree spectrum, i.e. the set of Turing degrees that compute an isomorphic copy of the structure. In this note we initiate a discussion of similar notions for topological structures, in particular we describe the degree spectra of domains.