  In 2004 Rosas and Sagan asked whether there was a way to define a basis in
the algebra of symmetric functions in noncommuting variables, NCSym, having
properties analogous to the classical Schur functions. This was because they
had constructed a partial such set that was not a basis. We answer their
question by defining Schur functions in noncommuting variables using a
noncommutative analogue of the Jacobi-Trudi determinant. Our Schur functions in
NCSym map to classical Schur functions under commutation, and a subset of them
indexed by set partitions forms a basis for NCSym. Amongst other properties,
Schur functions in NCSym also satisfy a noncommutative analogue of the product
rule for classical Schur functions in terms of skew Schur functions.
  We also show how Schur functions in NCSym are related to Specht modules, and
naturally refine the Rosas-Sagan Schur functions. Moreover, by generalizing
Rosas-Sagan Schur functions to skew Schur functions in the natural way, we
prove noncommutative analogues of the Littlewood-Richardson rule and coproduct
rule for them. Finally, we relate our functions to noncommutative symmetric
functions by proving a subset of our functions are natural extensions of
noncommutative ribbon Schur functions, and immaculate functions indexed by
integer partitions.
