  We give a complete characterization of invariant subspaces for $(M_{z_1},
\ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$ over the unit polydisc
$\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In particular, this yields a complete
set of unitary invariants for invariant subspaces for $(M_{z_1}, \ldots,
M_{z_n})$ on $H^2(\mathbb{D}^n)$, $n > 1$. As a consequence, we classify a
large class of $n$-tuples, $n > 1$, of commuting isometries. All of our results
hold for vector-valued Hardy spaces over $\mathbb{D}^n$, $n > 1$. Our invariant
subspace theorem solves the well-known open problem on characterizations of
invariant subspaces of the Hardy space over the unit polydisc.
