  Let $U^{n}$ be the unit polydisc of ${\Bbb C}^{n}$ and $\phi=(\phi_1, >...,
\phi_n)$ a holomorphic self-map of $U^{n}.$ By ${\cal B}^p(U^{n})$, ${\cal
B}^p_{0}(U^{n})$ and ${\cal B}^p_{0*}(U^{n})$ denote the $p$-Bloch space,
Little $p$-Bloch space and Little star $p$-Bloch space in the unit polydisc
$U^n$ respectively, where $p, q>0$. This paper gives the estimates of the
essential norms of bounded composition operators $C_{\phi}$ induced by $\phi$
between ${\cal B}^p(U^n)$ (${\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$) and
${\cal B}^q(U^n)$ (${\cal B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$). As their
applications, some necessary and sufficient conditions for the bounded
composition operators $C_{\phi}$ to be compact from ${\cal B}^p(U^n)$ $({\cal
B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n))$ into ${\cal B}^q(U^n)$ (${\cal
B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$) are obtained.
