  In this paper, we are concerned with the monotonic and symmetric properties
of convex solutions to fully nonlinear elliptic systems. We mainly discuss
Monge-Amp\`ere type systems for instance, considering
  \begin{equation*}
  \det(D^2u^i)=f^i(x,{\bf u},\nabla u^i), \ 1\leq i\leq m,
  \end{equation*} over bounded domains of various cases, including the bounded
smooth simply connected domains and bounded tube shape domains in
$\mathbb{R}^n$. We obtain monotonic and symmetric properties of the solutions
to the problem with respect to the geometry of domains and the monotonic and
symmetric properties of right-hand side terms. The proof is based on carefully
using the moving plane method together with various maximum principles and
Hopf's lemmas. The existence and uniqueness to an interesting example of such
system is also discussed as an application of our results.
