Consider the degenerate Fisher equation in one dimensional space or in a cylinder R×Ω,with bounded Ω(∈)Rn, {ut =△u + β(y)ux +up(1-u),t ＞0,(x,y)∈R×Ω,(a)u/(a)m=0,t ＞0,(x,y)∈R×(a)Ω,with p＞1.(1)It is well known that in one dimensional space there exists a minimal speed c*(p)＞0 such that (1)has a traveling front solution Uc(x-ct)satisfying U(-∞)= 1 and U(+∞)= 0 if and only if c≥c*(p); and the wave front Uc(z)with the minimal speed c=c*(p)decay exponentially at z=－∞ and is globally exponentially stable in exponentially weighted spaces,while the waves with noncritical speeds decay algebraically at z=+∞ and is locally stable in some algebraically weighted spaces.In this talk we shall be more interested in the existence of generalized traveling waves and the asymptotic behavior of solutions of(1)in one dimensional space with more general initial values decaying non-exponentially in space.Our results show that for more general slowly decaying initial value the solution still moves like a wave front and the decaying rate of initial value uniquely determines the asymptotic speed of the level set of the solution.In this talk we shall also talk about our recent progress on the existence and stability of cylinder waves of(1)in higher dimensional space when Its a joint work with Yanxia Wu and Junfeng He.