We discuss a relatively new connection between algebraic geometry and convex geometry. We explain a basic construction which associates convex bodies to semigroups of integral points. We see how this gives rise to convex bodies associated to algebraic varieties encoding information about their geometry. This far generalizes the notion of Newton polytope of a Laurent polynomial/toric variety. As an application, one gets a formula for the number of solutions of an algebraic system of equations on any variety, in terms of volumes of these bodies, far generalizing the well-known BKK theorem. Time permitting we will mention some interesting applications in algebraic geometry and symplectic geometry. There is also a local version of this theory concerning multiplicities of ideals. For the most part, the talk should be accessible to anybody with some background in algebra and geometry.