This phenomenon of self-replicating spots and pulses was discovered in the Gray-Scott model and in a series of gel reactor experiments. In this talk, I will present some analytical and numerical results for pulse dynamics and self-replication in one and two space dimensions in the Gray-Scott model and a class of activator-inhibitor systems. The results have been obtained in collaboration with Arjen Doelman, Wiktor Eckhaus, Rob Gardner, Dave Morgan, and Bert Peletier.