We investigate systematically into the various possible notions of traceable sets and the relations they bear to each other and to other notions such as diagonally noncomputable sets or complex and autocomplex sets. We review known notions and results that appear in the literature in different contexts, put them into perspective and provide simplified or at least more direct proofs. In addition, we introduce notions of traceability and complexity such as infinitely often versions of jump traceability and of complexity, and derive results about these notions that partially can be viewed as a natural completion of the previously known results. Finally, we give a result about polynomial-time bounded notions of traceability and complexity that shows that in principle the equivalences derived so far can be transferred to the time-bounded setting.