Conjugator length in finitely presented groups

The conjugator length function of a finitely generated group $G$ gives the minimal upper bound on the length of a conjugator for a pair of words that represent conjugate elements in $G$, as a function of the sum of the lengths of the words. Here, we seek to promote the systematic study of conjugator length functions by explaining their significance, by surveying what is known about them and by explaining fundamental techniques and examples.