  In this work, we consider the inhomogeneous nonlinear Schr\"odinger (INLS)
equation in $\mathbb{R}^n$ \begin{align} i\partial_t u + \Delta u + \gamma
|x|^{-b}|u|^{\alpha} u = 0, \end{align} where $\gamma=\pm 1$, and $\alpha$ and
$b$ are positive numbers. Our main focus is to estabilish the global
well-posedness of the INLS equation in Lorentz spaces for $0<b<2$ and
$\alpha<\frac{4-2b}{N-2}$. To achieve this, we use Strichartz estimates in
Lorentz spaces $L^{r,q}(\R^n)$ combined with a fixed point argument. Working on
Lorentz space setting instead the classical $L^p$ is motivated by the fact that
the potential $|x|^{-b}$ does not belong the usual $L^p$-space. As a
consequence of the ideas developed here on the global solution study we obtain
some other properties for INLS, such as, existence of self-similar solutions,
scattering, wave operators and assymptotic stability.
