  We introduce a new basis for quasisymmetric functions, which arise from a
specialization of nonsymmetric Macdonald polynomials to standard bases, also
known as Demazure atoms. Our new basis is called the basis of quasisymmetric
Schur functions, since the basis elements refine Schur functions in a natural
way. We derive expansions for quasisymmetric Schur functions in terms of
monomial and fundamental quasisymmetric functions, which give rise to
quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux.
  From here we derive a Pieri rule for quasisymmetric Schur functions that
naturally refines the Pieri rule for Schur functions. After surveying
combinatorial formulas for Macdonald polynomials, including an expansion of
Macdonald polynomials into fundamental quasisymmetric functions, we show how
some of our results can be extended to include the $t$ parameter from
Hall-Littlewood theory.
