It is difficult to imagine mathematics or engineering to date without nonlinear algebraic systems; these topics have deserved more diverse and vigorous interest over the years. In fact, one of the most important modern branches of mathematics originated in this context, topology, which has been inseparable from homotopy theory, describes a powerful intuitive tool to solve nonlinear algebraic systems. In the present work, we attempt to briefly summarize the theoretical beginnings of homotopy, its eventual transformation into a numerical method, its most reiterative applications in chemical engineering, and some current challenges in this field from an engineering-friendly perspective, especially approachable for chemical engineers.