  Let $0<\alpha<2$ be any real number. In this paper, we investigate the
following semilinear equations involving the fractional Laplacian
\begin{equation}(-\bigtriangleup)^{\alpha/2} u(x)=f(u),\end{equation} on
$\mathbb{R}^n$ and $\mathbb{R}^n_+$.
  Applying a direct method of moving planes for the fractional Laplacian, we
prove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and
$\mathbb{R}^n_+$ under mild conditions on $f$.
