For a natural number n, write sigma(n) for the sum of the positive divisors of n. The ancient Greeks called n deficient, perfect, or abundant, depending on whether sigma(n)/n was smaller than, equal, or larger than 2 (respectively). In this talk we discuss some problems motivated by this ancient classification. Much of this talk will be a survey of existing results and methods, but in the last part of the talk we will discuss some new results on how close the fraction sigma(n)/n is to being in lowest terms.