The fundamental basis of the Hopf algebra of quasisymmetric functions, QSym, can be thought of in terms of shuffling permutations.  We can think of QSym as having a basis indexed by equivalence classes of permutations, where we identify permutations with the same descent set. This descent set, Des, is a simple example of a permutation statistic that exhibits a property called being shuffle compatible. We will show that permutation statistics that are shuffle compatible give rise to “shuffle algebras” that are quotients of QSym and then discuss some bijective proofs that certain statistics are also shuffle compatible.