Plants can generate a spectacular array of complex shapes, many of which exhibit elaborate curvature in three dimensions, illustrated for example by orchid flowers and pitcher-plant traps. All of these structures arise through differential growth. Recent findings provide fresh mechanistic insights into how regional cell behaviours may lead to tissue deformations, including anisotropies and curvatures, which shape growing volumes and sheets of cells. Here were review our current understanding of how genes, growth, mechanics, and evolution interact to generate diverse structures. We illustrate problems and approaches with the complex three-dimensional trap of the bladderwort, Utricularia gibba, to show how a multidisciplinary approach can be extended to new model systems to understand how diverse plant shapes can develop and evolve.