A central problem in developmental biology is to understand how tissues are patterned in time and space - how do identical cells differentiate to form the adult body plan? Patterns often arise from prior asymmetries in developing embryos, but there is also increasing evidence for self-organizing mechanisms that can break the symmetry of an initially homogeneous cell population. I will present our recent experimental and theoretical analyses of how biophysical properties such as signal diffusion and stability contribute to axis formation and tissue allocation during vertebrate embryogenesis, and how we can use these insights to engineer self-organizing synthetic multicellular patterning systems.