  In this paper, we develop a systematical approach in applying an asymptotic
method of moving planes to investigate qualitative properties of positive
solutions for fractional parabolic equations. We first obtain a series of
needed key ingredients such as narrow region principles, and various asymptotic
maximum and strong maximum principles for antisymmetric functions in both
bounded and unbounded domains. Then we illustrate how this new method can be
employed to obtain asymptotic radial symmetry and monotonicity of positive
solutions in a unit ball and on the whole space. Namely, we show that no matter
what the initial data are, the solutions will eventually approach to radially
symmetric functions.
  We firmly believe that the ideas and methods introduced here can be
conveniently applied to study a variety of nonlocal parabolic problems with
more general operators and more general nonlinearities.
