In this talk,  I will discuss the existence and asymptotic behavior of traveling wave solutions to Allen-Cahn
equation with fractional Laplacians where the double well potenotial has unequal depths.   A key ingredient is
the estimate of the speed of the traveling wave in terms of the potential, which seems new even for the classical
Allen-Cahn equation.  I will also discuss nonexistence of traveling wave solutions to a nonlocal  combustion model.
The talk is based on recent results obtained jointly with Tingting Huan and with  Mingfeng Zhao respectively.