  In the Gilbert-Shannon-Reeds shuffle, a deck of $N$ cards is cut into two
approximately equal parts which are then riffled uniformly at random. Bayer and
Diaconis famously showed that this Markov chain undergoes cutoff in total
variation after $\frac{3\log(N)}{2 \log(2)}$ shuffles. We establish cutoff for
the more general asymmetric riffle shuffles in which one cuts the deck into
differently sized parts before riffling. The value of the cutoff point confirms
a conjecture of Lalley from 2000. Some appealing consequences are that
asymmetry always slows mixing and that total variation mixing is strictly
faster than separation and $L^{\infty}$ mixing.
