An important issue in the study of reaction-diffusion equations is determining whether or not solutions blow-up. This carries over to numerical solutions, in which case we have the additional concern of determining if discretization either inhibits blowup or causes it to occur artificially. The existence of invariant rectangles inside of which solutions remain for all time is an important factor for addressing these issues in many cases. After presenting some motivating examples, we discuss the preservation of invariant rectangles under discretization in two ways. We construct special numerical methods that preserve invariant rectangles exactly and show how to use adaptive error control to preserve invariant rectangles in an approximate sense.