In this talk, we shall survey some recent results on the wave propagation in the two species competition systems with Lotka-Volterra type nonlinearity. This includes systems with continuous and discrete diffusion. Both monostable and bistable cases shall be discussed. Questions on minimal wave speed for the monostable case, propagation failure in the bistable case, monotonicity of wave profiles, and uniqueness of wave speed in the bistable case shall be addressed.