Abstract: Informally, if two functions can be “continuously deformed” from one to the other, this is called a homotopy. Homotopy emerged from theory more than a century ago and was introduced as a numerical method for solving non-linear problems in the 1960s. The basic components of these early continuation algorithms built on earlier path-tracking methods, which exist in today’s “black box” solvers. We will look at the building blocks of continuation algorithms, how they work (or don’t), and how key insights from the 1970s and 1980s contributed to some powerful polynomial-solving software packages.