The unknotting number u(K) of a knot K is the minimal number of times you must allow K to pass through itself in order to unknot it. Although this is one of the oldest and most natural knot invariants, it remains mysterious. We will survey known results on u(K), including some recent joint work with John Luecke on when so-called algebraic knots have u(K) = 1, and discuss several open questions.