abstract: The homotopy method is one of the most succesful ones in finding solutions to systems of polynomial equations. Given a target system, one considers another system whose solution is known and a smooth curve (even just a segment) in the space of systems joining both systems. Then, the known solution is tracked along the path to approximate a solution of the target system. A fundamental question is "where to start the homotopy?" In this talk I will discuss some history, some recent results and some open problems in answering this question.