  In the present paper, we consider the coupled Schr\"{o}dinger systems with
critical exponent: \begin{equation*} \begin{cases} -\Delta
u_i+\lambda_{i}u_i=\sum\limits_{j=1}^{d} \beta_{ij}|u_j|^{3}|u_i|u_i \quad
~\text{ in } \Omega,\\ u_i \in H_0^1(\Omega) ,\quad i= 1,2,...,d. \end{cases}
\end{equation*} Here, $\Omega\subset \mathbb{R}^{3}$ is a smooth bounded
domain, $d \geq 2$, $\beta_{ii}>0$ for every $i$, and $\beta_{ij}=\beta_{ji}$
for $i \neq j$. We study a Br\'{e}zis-Nirenberg type problem:
$-\lambda_{1}(\Omega)<\lambda_{1},\cdots,\lambda_{d}<-\lambda^*(\Omega)$, where
$\lambda_{1}(\Omega)$ is the first eigenvalue of $-\Delta$ with Dirichlet
boundary conditions and $\lambda^*(\Omega)\in (0, \lambda_1(\Omega))$. We
acquire the existence of least energy positive solutions to this system for
weakly cooperative case ($\beta_{ij}>0$ small) and for purely competitive case
($\beta_{ij}\leq 0$) by variational arguments. The proof is performed by
mathematical induction on the number of equations, and requires more refined
energy estimates for this system. Besides, we present a new nonexistence
result, revealing some different phenomena comparing with the
higher-dimensional case $N\geq 5$. It seems that this is the first paper to
give a rather complete picture for the existence of least energy positive
solutions to critical Schr\"{o}dinger system in dimension three.
