What can topos theory do for algebra and geometry?

Toposes are special kinds of categories. Because of a device called "internal language", they can be regarded as alternate mathematical universes in which not the usual laws of logic hold. Various subjects such as differential geometry, algebraic geometry, homotopy theory, commutative algebra, measure theory and computability theory provide sources for such toposes, where they yield a way to view the familiar objects of study from a new point of view. The talk gives an introduction to this circle of ideas and surveys some of their applications, focusing on reflection principles and local-to-global principles in geometry and new reduction techniques in commutative algebra. No prior knowledge of category theory, topos theory or formal logic is supposed.