  We prove the existence of solutions $(\lambda, v)\in \mathbb{R}\times
H^{1}(\Omega)$ of the elliptic problem \[ \begin{cases}
  -\Delta v+(V(x)+\lambda) v =v^{p}\ &\text{ in $ \Omega, $}
  \ v>0,\qquad \int_\Omega v^2\,dx =\rho. \end{cases} \] Any $v$ solving such
problem (for some $\lambda$) is called a normalized solution, where the
normalization is settled in $L^2(\Omega)$. Here $\Omega$ is either the whole
space $\mathbb R^N$ or a bounded smooth domain of $\mathbb R^N$, in which case
we assume $V\equiv0$ and homogeneous Dirichlet or Neumann boundary conditions.
Moreover, $1<p<\frac{N+2}{N-2}$ if $N\ge 3$ and $p>1$ if $N=1,2$. Normalized
solutions appear in different contexts, such as the study of the Nonlinear
Schr\"odinger equation, or that of quadratic ergodic Mean Field Games systems.
We prove the existence of solutions concentrating at suitable points of
$\Omega$ as the prescribed mass $\rho$ is either small (when $p<1+\frac 4N$) or
large (when $p>1+\frac 4N$) or it approaches some critical threshold (when
$p=1+\frac 4N$).
