We review a chemotaxis system with flux limitation and present known results for both: the elliptic and the parabolic-elliptic case. We denote by u and v the density of living organisms and the concentration of a chemical substance, respectively, \( \left\{ \begin{array}{ll} - \ div(A(x) \nabla u) + u = -div(u D(x) \, |\nabla v|^{p-1} \nabla v ) + f(x), \; \; \text{ in } x\in \Omega , \\[2mm] - \ div(D(x) \nabla v) +v= u^{\theta }, \; \; \text{ in } x\in \Omega , \\[2mm] u(x)=v(x)=0, \; \; \text{ in } x\in \partial \Omega , \end{array} \right. \) for given A, D and f, under some restrictions on \(p \in (1,2)\) and \(\theta \in (0,1)\) ; \( \left\{ \begin{array}{l} \displaystyle u_t-\Delta u= - div (\chi u|\nabla v|^{p-2}\nabla v), \quad \quad x\in \Omega ,\quad t>0, \\[2mm] \displaystyle -\Delta v = u-\frac{1}{|\Omega |} \int _{\Omega } u_0 dx,\quad \quad x\in \Omega ,\quad t>0, \\[2mm] \displaystyle \frac{\partial u}{\partial n }= \frac{\partial u}{\partial n}=0, \quad \;\; \;\;x\in \partial \Omega ,\quad t>0, \\[2mm] u(0,x)= u_0(x), \quad x\in \Omega , \end{array} \right. \) where \(p \in (1,2)\) . For \(p<\frac{N}{N-1}\) ( \(N \ge 2\) ), the solutions of the parabolic elliptic system globally exists on time and for \(p\in (N/(N-1),2)\) , there exists initial data, such that, the solutions blow up in finite time. The details of the results can be found in Boccardo and Tello AML 2022; Negreanu and Tello JDE 2018 and Tello CPDE 2021.