Probabilistic Diophantine approximation is a branch of analytic number theory which studies how well typical real numbers (in the sense of Lebesgue measure) can be approximated by rational numbers with small denominators. A long standing central problem in the subject was a conjecture, attributed to a 1941 paper of Duffin and Schaeffer, which was proved last year by Koukoulopoulos and Maynard. In this talk we will go over some of the history and motivation for this problem, and we will explain the basic probabilistic ideas and machinery which has been developed over the years for attacking it. This lecture will be aimed primarily at graduate students, and is intended to provide some background for the upcoming colloquium (January 22) on the proof of the Duffin-Schaeffer conjecture.