We study two aspects of the elasticity of smectic-A elastomers that make these materials genuinely and qualitatively different from conventional uniaxial rubbers. Under strain applied parallel to the layer normal, monodomain smectic-A elastomers exhibit a drastic change in Young’s modulus above a threshold strain value of about 3%, as has been measured in experiments by [Nishikawa and Finkelmann, Macromol. Chem. Phys. 200, 312 (1999)]. Our theory predicts that such strains induce a transition to a smectic-C-like state and that it is this transition that causes the change in elastic modulus. We calculate the stress-strain behavior as well as the tilt of the smectic layers and the molecular orientation for strain along the layer normal, and we compare our findings with the experimental data. We also study the electroclinic effect in chiral smectic-A* elastomers. According to experiments by [Lehmann et al., Nature (London) 410, 447 (2001)] and [Köhler et al., Appl. Phys. A 80, 381 (2003)], this effect leads in smectic-A* elastomers to a giant or, respectively, at least very large lateral electrostriction. Incorporating polarization into our theory, we calculate the height change of smectic-A* elastomer films in response to a lateral external electric field, and we compare this result to the experimental findings.