Cutoff is a remarkable property of many Markov chains in which they rapidly transition from an unmixed to a mixed state. Most random walks on the symmetric group, also known as card shuffles, are believed to mix with cutoff, but we are far from being able to proof this. We will survey existing cutoff results and techniques for random walks on the symmetric group, and present three recent results: cutoff for a biased transposition walk, cutoff for the random-to-random card shuffle (answering a 2001 conjecture of Diaconis), and pre-cutoff for the involution walk. The results use either probabilistic techniques such as strong stationary times or diagonalization through algebraic combinatorics and representation theory of the symmetric group.