  A d-contraction is a d-tuple $(T_1,...,T_d)$ of mutually commuting operators
acting on a common Hilbert space H such that $ \|T_1\xi_1+T_2\xi_2+...
+T_d\xi_d\|^2\leq \|\xi_1\|^2+\|\xi_2\|^2+...+\|\xi_d\|^2 $ for all
$\xi_1,\xi_2,...,\xi_d\in H$. These are the higher dimensional counterparts of
contractions. We show that many of the operator-theoretic aspects of function
theory in the unit disk generalize to the unit ball B_d in complex d-space,
including von Neumann's inequality and the model theory of contractions. These
results depend on properties of the d-shift, a distinguished d-contraction
which acts on a new $H^2$ space associated with B_d, and which is the higher
dimensional counterpart of the unilateral shift. $H^2$ and the d-shift are
highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the
d-shift relative to its generated C^*-algebra we find that there is more
uniqueness in dimension $d\geq 2$ than there is in dimension one.
