  A fixing set $\mathcal{F}$ of a graph $G$ is a set of those vertices of the
graph $G$ which when assigned distinct labels removes all the automorphisms
from the graph except the trivial one. The fixing number of a graph $G$,
denoted by $fix(G)$, is the smallest cardinality of a fixing set of $G$. In
this paper, we study the fixing number of composition product, $G_1[G_2]$ and
corona product, $G_1 \odot G_2$ of two graphs $G_1$ and $G_2$ with orders $m$
and $n$ respectively. We show that for a connected graph $G_1$ and an arbitrary
graph $G_2$ having $l\geq 1$ components $G_2^1$, $G_2^2$, ... $G_2^l,$
$mn-1\geq fix(G_1[G_2])\geq m\left(\sum \limits_{i=1}^{l} fix(G_2^i )\right)$.
For a connected graph $G_1$ and an arbitrary graph $G_2$, which are not
asymmetric, we prove that $fix(G_1\odot G_2)=m fix( G_2)$. Further, for an
arbitrary connected graph $G_{1}$ and an arbitrary graph $G_{2}$ we show that
$fix(G_1\odot G_2)= max\{fix(G_1), m fix(G_2)\}$.
