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.