Finding dc operating points, steady state, and transient responses of electronic circuits are essential tasks in electrical circuit simulation and involve solving nonlinear differential/algebraic equations. Traditional methods for solving such systems of equations often fail, are difficult to converge, and, often cannot find all the solutions. We investigate the application of homotopy methods to solving nonlinear equations describing circuits consisting of bipolar junction and MOS transistors that traditionally pose simulation difficulties. In this talk, I will describe experiments with homotopies that led to better understanding of homotopy algorithms and the behavior of nonlinear circuits, and, ultimately, to the development of better circuit simulation tools.