Abstract: Hilbert asked for an algorithm to determine whether an arbitrary polynomial equation (several variables) with integers coefficients has a solution (in the integers). Through the cumulative work of many people (Turing, Davis, Putnam, Robinson, Matiyasevich), it is known that no such algorithm exists. The analogue of Hilbert's problem for rings other than the integers is an active area of research. I'll try to give an indication of how the problem for the integers was resolved. I'll tell you a little too about other rings, including a recent breakthrough.