The general subject of this talk will be the question of whether an object (such as a cake or piece of land) can be divided among a number of people so that each receives a portion he considers a fair share, according to his own values. Formally, there are n measures on the same object - a measurable space - and a typical goal is to find a partition of the object into n (measurable) pieces so that the minimum value of the i-th measure on the i-th piece is as large as possible. Classical problems such as Steinhaus' "Cake-cutting Problem" and "Ham Sandwich Problem", Neyman and Pearson's "Bisection Problem", and Fisher's "Problem of the Nile" will be mentioned, and several results based on generalizations of Lyapounov's Convexity Theorem will be discussed, along with applications to disarmament, dividing inheritances, and selection of college deans or department chairs. The talk will be aimed for the non-specialist.