  We consider the Lane-Emden Dirichlet problem \begin{equation}\tag{1}
\left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }\Omega
u=0\qquad\qquad\qquad\mbox{ on }\partial \Omega \end{array}\right.
\end{equation} when $p>1$ and $\Omega\subset\mathbb R^2$ is a smooth bounded
domain. The aim of the paper is to survey some recent results on the asymptotic
behavior of solutions of (1) as the exponent $p\rightarrow \infty $.
