This is the first of two talks. We will sketch Rips and Sela's construction of canonical representatives for hyperbolic groups and show why they allow solving equations over these groups. Rips and Sela asked whether hyperbolic groups admit stable canonical representatives. In this talk we will explain what stability means, and in a sequel to this talk we will discuss existence of stable canonical representatives in certain hyperbolic groups, namely, geometric small cancelation groups. This is joint work with Yasushi Yamashita (Nara Women's University)