Algebraic K-theory is a deep and subtle invariant of rings and schemes, carrying information about arithmetic and geometry. When applied to the group ring of the loop space of a manifold, it captures information about the diffeomorphism group. Over the past 25 years, the study of algebraic K-theory has been revolutionized by the introduction of trace methods, which use trace (or Chern character) maps to the simpler but related theories of (topological) cyclic and Hochschild homology. A unifying perspective on the properties of algebraic K-theory and these related theories is afforded by viewing the input as a category of compact modules (i.e., a piece of an enhanced triangulated category). This talk will survey recent work describing the structural properties of these theories using various models of the homotopical category of module categories.