Knots and links are fascinating objects to study as an exercise in geometry and topology, and they have surprising applications across the sciences. In my view, however, the "best" way to understand them comes through abstract algebra. In this talk I'll attempt to show how knot theory naturally gives rise to algebraic structures, and in particular one structure -- the quandle -- which was tailor-made to classify oriented knots. We'll see how these structures variously address the most fundamental problem in knot theory: how to determine whether two knots are the same knot (or not!).