The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian ${/rm Gr}$ associated with $SL(n, /mathbb{C})$.Their main results are: Pieri rules for the Schubert bases of $H^*({/rm Gr})$ and $H_*({/rm Gr})$, which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. A new combinatorial definition for $k$-Schur functions, which represent the Schubert basis of $H_*({/rm Gr})$. A combinatorial interpretation of the pairing $H^*({/rm Gr})/times H_*({/rm Gr}) /rightarrow/mathbb Z$ induced by the cap product.