  We introduce and investigate $H^\infty$-functional calculus for commuting
finite families of Ritt operators on Banach space $X$. We show that if either
$X$ is a Banach lattice or $X$ or $X^*$ has property $(\alpha)$, then a
commuting $d$-tuple $(T_1,\ldots, T_d)$ of Ritt operators on $X$ has an
$H^\infty$ joint functional calculus if and only if each $T_k$ admits an
$H^\infty$ functional calculus. Next for $p\in(1,\infty)$, we characterize
commuting $d$-tuple of Ritt operators on $L^p(\Omega)$ which admit an
$H^\infty$ joint functional calculus, by a joint dilation property. We also
obtain a similar characterisation for operators acting on a UMD Banach space
with property $(\alpha)$. Then we study commuting $d$-tuples $(T_1,\ldots,
T_d)$ of Ritt operators on Hilbert space. In particular we show that if $\Vert
T_k\Vert\leq 1$ for every $k=1,\ldots,d$, then $(T_1,\ldots, T_d)$ satisfies a
multivariable analogue of von Neumann's inequality. Further we show analogues
of most of the above results for commuting finite families of sectorial
operators.
