The conjugacy problem for a finitely generated group G asks for an algorithm which, on input a pair of words u and v, declares whether or not they represent conjugate elements of G. The conjugator length function CL is its most direct quantification: CL(n) is the minimal N such that if u and v represent conjugate elements of G and the sum of their lengths is at most n, then there is a word w of length at most N such that uw=wv in G. I will talk about why this function is interesting and how it can behave, and I will highlight some open questions. En route I will talk about results variously with Martin Bridson, Conan Gillis, and Andrew Sale, as well as recent advances by Conan Gillis and Francis Wagner.