The non-commutative symmetric functions and quasi-symmetric functions are the second and third examples of a combinatorial Hopf algebra that one encounters (the first being the symmetric functions). In recent years there have been at least two bases proposed as analogues of the Schur functions and they are in addition to the "ribbon=fundamental^*" basis. I´ll list properties that we would want these bases to have as analogues of the Schur functions and then explain some computational results that tell us what is possible (surprisingly, it is not possible to have it all!). I will also discuss some symmetric function positivity open problems that we hope these bases will resolve.

This is joint work with Laura Colmenarejo.