it has been observed since the early work of richard stanley that several well-known permutation statistics are ``compatible'' with the operation of shuffling permutations. in joint work with ira gessel, we formalize this notion of a shuffle-compatible permutation statistic and develop a unifying framework for studying shuffle-compatibility, which has close connections to the theory of p-partitions, quasisymmetric functions, and noncommutative symmetric functions. in this talk, i will survey the main results of our work as well as several new directions of research concerning shuffle-compatibility.