Abstract I present new techniques and results concerning the stability of travelling waves to semilinear parabolic equations. I am concerned with those problems in which the essential spectrum of the linearization about the wave contains λ = 0. In these circumstances, the stability results of D. Henry ( in "Lecture Notes in Mathematics," Vol. 840, Springer-Verlag, New York, 1981) and P. Bates and C. Jones ( Dynamics Rep. 2 (1989), 1-38) are not applicable, as the spectral information does not yield an exponential time decay estimate on the semigroup. I show that under certain (relatively) easily computable conditions the wave is stable in polynomially weighted L ∞ spaces. Furthermore, the rate at which the perturbation decays to the wave as t → ∞ depends on the growth rate of the polynomial near infinity. The method of proof uses the semigroup of the linearized equation, with solutions of the full problem expressed by the variation of constants formula. Estimates for the semigroup are derived through a new technique for estimating the resolvent.