Shape-morphing structures are at the core of future applications in aeronautics(1), minimally invasive surgery(2), tissue engineering(3) and smart materials(4). However, current engineering technologies, based on inhomogeneous actuation across the thickness of slender structures, are intrinsically limited to one-directional bending(5). Here, we describe a strategy where mesostructured elastomer plates undergo fast, controllable and complex shape transformations under applied pressure. Similar to pioneering techniques based on soft hydrogel swelling(6-10), these pneumatic shape-morphing elastomers, termed here as 'baromorphs', are inspired by the morphogenesis of biological structures(11-15). Geometric restrictions are overcome by controlling precisely the local growth rate and direction through a specific network of airways embedded inside the rubber plate. We show how arbitrary three-dimensional shapes can be programmed using an analytic theoretical model, propose a direct geometric solution to the inverse problem, and illustrate the versatility of the technique with a collection of configurations.