The development of multicellular organisms relies on a small set of construction techniques—assembly, sculpting, and folding—that are spatially and temporally regulated in a combinatorial manner to produce the diversity of tissues within the body. These basic processes are well conserved across tissue types and species at the level of both genes and mechanisms. Here we review the signaling, patterning, and biomechanical transformations that occur in two well-studied model systems of epithelial folding to illustrate both the complexity and modularity of tissue development. In particular, we discuss the possibility of a spatial code specifying morphogenesis. To decipher this code, engineers and scientists need to establish quantitative experimental systems and to develop models that address mechanisms at multiple levels of organization, from gene sequence to tissue biomechanics. In turn, quantitative models of embryogenesis can inspire novel methods for creating synthetic organs and treating degenerative tissue diseases.