The conjugator length function of a finitely generated group is the function f so that f(n) is the minimal upper bound on the length of a word realizing the conjugacy of two words of length at most n. This function provides a measure for the complexity of a direct approach to the Conjugacy Problem for the finitely generated group. I will discuss what functions can be realized as the conjugator length function of a finitely presented group and the connection of this function with other important invariants of finitely presented groups.