  We investigate normalized solutions for the Schr\"{o}dinger equation with
combined Hartree type and power nonlinearities, namely \begin{equation*}
\left\{ \begin{array}{ll} -\Delta u+\lambda u=\gamma (I_{\alpha }\ast
\left\vert u\right\vert ^{p})|u|^{p-2}u+\mu |u|^{q-2}u & \quad \text{in}\quad
\mathbb{R}^{N}, \\ \int_{\mathbb{R}^{N}}|u|^{2}dx=c, & \end{array}% \right.
\end{equation*} where $N\geq 2$ and $c>0$ is a given real number. Under
different assumptions on $\gamma ,\mu ,p$ and $q$, we prove several
nonexistence, existence and multiplicity results. In particular, we are more
interested in the cases when the competing effect of Hartree type and power
nonlinearities happens, i.e. $\gamma \mu <0,$ including the cases $\gamma
<0,\mu >0$ and $% \gamma >0,\mu <0.$ Due to the different "strength" of two
types of nonlinearities, we find some differences in results and in the
geometry of the corresponding functionals between these two cases.
