Differential equations containing multiple, widely-separated timescales are termed "stiff". Textbooks routinely state that implicit methods must be used to solve such equations because stability limits on the timestep size make explicit integration completely impractical. In direct contradiction to this prevailing view in the literature, this talk will show that even extremely stiff sets of ordinary differential equations may be solved efficiently by explicit methods if limiting algebraic solutions are used to stabilize the numerical integration. Using stringent tests with astrophysical thermonuclear networks, evidence is provided that these methods can deal with the stiffest networks, even in the approach to equilibrium, with accuracy and integration timestepping comparable to that of standard implicit methods. Explicit algorithms can execute a timestep faster and scale more favorably with network size than implicit algorithms. Thus, these results suggest that algebraically-stabilized explicit methods might enable integration of larger reaction networks coupled to fluid dynamics than has been feasible previously for many applications in astrophysics and a variety of other disciplines.