  The random-to-top and the riffle shuffle are two well-studied methods for
shuffling a deck of cards. These correspond to the symmetric group $S_n$, i.e.,
the Coxeter group of type $A_{n-1}$. In this paper, we give analogous shuffles
for the Coxeter groups of type $B_n$ and $D_n$. These can be interpreted as
shuffles on a ``signed'' deck of cards. With these examples as motivation, we
abstract the notion of a shuffle algebra which captures the connection between
the algebraic structure of the shuffles and the geometry of the Coxeter groups.
We also briefly discuss the generalisation to buildings which leads to
q-analogues.
