Two species (designated by 0's and 1's) compete for territory on a lattice according to the rules of a voter model, except that the 0's jump d 0 spaces and the 1's jump d 1 spaces. When d 0 = d 1 = 1 the model is the usual voter model. It is shown that in one dimension, if d 1 > d 0 and d 0 = 1,2 and initially there are infinitely many blocks of 1's of length ≥ d 1, then the 1's eliminate the 0's. It is believed this may be true whenever d 1 > d 0. In the biased annihilating branching process particles place offspring on empty neighbouring sites at rate λ and neighbouring pairs of particles coalesce at rate 1. In one dimension it is known to converge to the product measure density λ/(1+λ) when λ ≥ 1/3, and the initial configuration is non-zero and finite. This result is extended to λ ≥ 0.0347. Bounds on the edge-speed are given.