Multiplicities of algebras have their roots in intersection theory in the first half of the 20th century with the work of several authors including Hilbert, Noether, and Van der Waerden. In the 1950s developments by Samuel, Serre, and Rees brought multiplicities to the interest of commutative algebraists. Since then multiplicities has become a very active research theme in commutative algebra with connections with other fields such as algebraic combinatorics, convex geometry, and singularity theory. In this talk I will survey the history and some results in this topic, and will include some open problems along the way.