  Maximality of a contractive tuple of operators is considered.
Characterization of a contractive tuple to be maximal is obtained. Notion of
maximality of a submodule of Drury-Arveson module on the $d$-dimensional unit
ball $\mathbb{B}_d$ is defined. For $d=1$, it is shown that every submodule of
the Hardy module over the unit disc is maximal. But for $d\ge 2$ we prove that
any homogeneous submodule or submodule generated by polynomials is not maximal.
A characterization of a submodule to be maximal is obtained.
