We study the rheological behavior of mixtures of foams and pastes, which can be described as suspensions of bubbles in yield stress fluids. Model systems are designed by mixing monodisperse aqueous foams and concentrated emulsions. The elastic modulus of the suspensions decreases with the bubble volume fraction. This decrease is all the sharper as the elastic capillary number (defined as the ratio of the paste elastic modulus to the bubble capillary pressure) is high, which accounts for the softening of the bubbles as compared to the paste. By contrast, the yield stress of most studied materials is not modified by the presence of bubbles. Their plastic behavior is governed by the plastic capillary number, defined as the ratio of the paste yield stress to the bubble capillary pressure. At low plastic capillary number values, bubbles behave as nondeformable inclusions, and we predict that the suspension dimensionless yield stress should remain close to unity. At large plastic capillary number values, we observe bubble breakup during mixing: bubbles are deformed by shear. Finally, at the highest bubble volume fractions, the yield stress increases abruptly: this is interpreted as a 'foamy yield stress fluid' regime, which takes place when the paste mesoscopic constitutive elements are strongly confined in the films between the bubbles.