In the present paper, we prove the existence of solutions $(\lambda_1,\lambda_2,u,v)\in\R^2\times H^1(\R^3,\R^2)$ to systems of coupled Schr\"odinger equations
$$
\begin{cases}
-\Delta u+\lambda_1u=\mu_1 u^3+\beta uv^2\quad &\hbox{in}\;\R^3\\
-\Delta v+\lambda_2v=\mu_2 v^3+\beta u^2v\quad&\hbox{in}\;\R^3\\
u,v>0&\hbox{in}\;\R^3
\end{cases}
$$
satisfying the normalization constraint
$
\displaystyle\int_{\R^3}u^2=a^2\quad\hbox{and}\;\int_{\R^3}v^2=b^2,
$
which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics.
The parameters $\mu_1,\mu_2,\beta>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the case of fixed frequencies $\lambda_1,\lambda_2$. When the masses are prescribed, the standard approach to this problem is variational with $\lambda_1,\lambda_2$ appearing as Lagrange multipliers. Here we present a new approach based on the fixed point index in cones, bifurcation theory, and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $\beta$ in a large range. We also have a result about the nonexistence of positive solutions which shows that our existence theorem is almost optimal. Especially, if $\mu_1=\mu_2$ we prove that normalized solutions exist for all $\beta>0$ and all $a,b>0$.