We consider the model of a false vacuum bubble with a thin wall where the surface energy density is composed of two different components, domain-wall type and dust type, with opposite signs. We find stably oscillating solutions, which we call breathing bubbles. By decay to a lower mass state, such a breathing bubble could become either i) a child universe or ii) a bubble that eats up the original universe, depending on the sign of the surface energy of the domain-wall component. We also discuss the effect of the finite-thickness corrections to the thin-wall approximation and possible origins of the energy contents of our model.