Let R -> A be a map of rings (or ring spectra) and suppose that A is finitely generated projective (or perfect) as an R-module. Then there is a wrong-way "transfer" map on algebraic K-theory, K(A) -> K(R). In particular, when E -> B is a fibration whose fiber F is a finite CW complex, this gives a wrong-way map on Waldhausen's functor A(B) -> A(E). We will ask a few fundamental questions about this transfer, and present the beginning of a program to answer these questions using trace methods. Our main results concern the corresponding transfer in topological Hochschild homology (THH), which is a stable map of free loop spaces LB -> LE. Aside from the insights into algebraic K-theory, these results give us apparently novel computational tools for the homology of free loop spaces. This is joint work with John Lind.