Throughout biology, function is intimately linked with form. Across scales ranging from subcellular to multiorganismal, the identity and organization of a biological structure's subunits dictate its properties. The field of molecular morphogenesis has traditionally been concerned with describing these links, decoding the molecular mechanisms that give rise to the shape and structure of cells, tissues, organs, and organisms. Recent advances in synthetic biology promise unprecedented control over these molecular mechanisms; this opens the path to not just probing morphogenesis but directing it. This review explores several frontiers in the nascent field of synthetic morphogenesis, including programmable tissues and organs, synthetic biomaterials and programmable matter, and engineering complex morphogenic systems de novo. We will discuss each frontier's objectives, current approaches, constraints and challenges, and future potential.