(Doctoral thesis defense) We introduce a new basis for QSym which we refer to as the quasisymmetric Schur functions and give combinatorial rules for them in terms of composition tableaux. We prove refinements of the classical Littlewood-Richardson rule that give interpretations for the expansion of the product of a symmetric Schur function with a quasisymmetric Schur function (resp. Demazure atom or character) in terms of the basis of quasisymmetric Schur functions (resp. Demazure atoms or characters).