  We define a new family of symmetric functions which are affine analogues of
Stanley symmetric functions. We establish basic properties of these functions
including symmetry, dominance and conjugation. We conjecture certain positivity
properties in terms of a subfamily of symmetric functions called affine Schur
functions. As applications, we show how affine Stanley symmetric functions
generalise the (dual of the) $k$-Schur functions of Lapointe, Lascoux and Morse
as well as the cylindric Schur functions of Postnikov. Conjecturally, affine
Stanley symmetric functions should be related to the cohomology of the affine
flag variety.
