  In this paper, we consider nonlinear equations involving the fractional
p-Laplacian $$ (-\lap)_p^s u(x)) \equiv C_{n,s,p} PV \int_{\mathbb{R}^n}
\frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-z|^{n+ps}} dz= f(x,u).$$
  We prove a {\em maximum principle for anti-symmetric functions} and obtain
other key ingredients for carrying on the method of moving planes, such as {\em
a key boundary estimate lemma}. Then we establish radial symmetry and
monotonicity for positive solutions to semilinear equations involving the
fractional p-Laplacian in a unit ball and in the whole space. We believe that
the methods developed here can be applied to a variety of problems involving
nonlinear nonlocal operators.
