  Recently a new basis for the Hopf algebra of quasisymmetric functions $QSym$,
called quasisymmetric Schur functions, has been introduced by Haglund, Luoto,
Mason, van Willigenburg. In this paper we extend the definition of
quasisymmetric Schur functions to introduce skew quasisymmetric Schur
functions. These functions include both classical skew Schur functions and
quasisymmetric Schur functions as examples, and give rise to a new poset
$\mathcal{L}_C$ that is analogous to Young's lattice. We also introduce a new
basis for the Hopf algebra of noncommutative symmetric functions $NSym$. This
basis of $NSym$ is dual to the basis of quasisymmetric Schur functions and its
elements are the pre-image of the Schur functions under the forgetful map
$\chi: NSym \rightarrow Sym$. We prove that the multiplicative structure
constants of the noncommutative Schur functions, equivalently the coefficients
of the skew quasisymmetric Schur functions when expanded in the quasisymmetric
Schur basis, are nonnegative integers, satisfying a Littlewood-Richardson rule
analogue that reduces to the classical Littlewood-Richardson rule under $\chi$.
  As an application we show that the morphism of algebras from the algebra of
Poirier-Reutenauer to $Sym$ factors through $NSym$. We also extend the
definition of Schur functions in noncommuting variables of Rosas-Sagan in the
algebra $NCSym$ to define quasisymmetric Schur functions in the algebra
$NCQSym$. We prove these latter functions refine the former and their
properties, and project onto quasisymmetric Schur functions under the forgetful
map. Lastly, we show that by suitably labeling $\mathcal{L}_C$, skew
quasisymmetric Schur functions arise in the theory of Pieri operators on
posets.
