  We consider the existence of \emph{normalized} solutions in $H^1(\R^N) \times
H^1(\R^N)$ for systems of nonlinear Schr\"odinger equations which appear in
models for binary mixtures of ultracold quantum gases. Making a solitary wave
ansatz one is led to coupled systems of elliptic equations of the form \[
\left\{ \begin{aligned} -\De u_1 &= \la_1u_1 + f_1(u_1)+\pa_1F(u_1,u_2),\\ -\De
u_2 &= \la_2u_2 + f_2(u_2)+\pa_2F(u_1,u_2),\\ u_1,u_2&\in H^1(\R^N),\ N\ge2,
\end{aligned} \right. \] and we are looking for solutions satisfying \[
\int_{\R^N}|u_1|^2 = a_1,\quad \int_{\R^N}|u_2|^2 = a_2 \] where $a_1>0$ and
$a_2>0$ are prescribed. In the system $\la_1$ and $\la_2$ are unknown and will
appear as Lagrange multipliers. We treat the case of homogeneous
nonlinearities, i.e.\ $f_i(u_i)=\mu_i|u_i|^{p_i-1}u_i$,
$F(u_1,u_2)=\be|u_1|^{r_1}|u_2|^{r_2}$, with positive constants $\be, \mu_i,
p_i, r_i$. The exponents are Sobolev subcritical but may be
$L^2$-supercritical:
$p_1,p_2,r_1+r_2\in]2,2^*[\,\setminus\left\{2+\frac4N\right\}$.
