The aim of this paper is to study the singular solutions to fractional elliptic equations with absorption $$ \left\{ \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+|u|^{p-1}u=0,\quad & \rm{in}\quad\Omega\setminus\{0\},\\[2mm] u=0,\quad & \rm{in}\quad \R^N\setminus\Omega,\\[2mm] \lim_{x\to 0}u(x)=+\infty, \end{array} \right. $$ where $p>0$, $\Omega$ is an open, bounded and smooth domain of $\R^N\ (N\ge2)$ with $0\in\Omega$. We analyze the existence, nonexistence, uniqueness and asymptotic behavior of the solutions.