Branching morphogenesis, the creation of branched structures in the body, is a key feature of animal and plant development. It requires the coordinated interplay of multiple types of epithelial cells with the surrounding extracellular matrix. Cell migration, proliferation, and extracellular matrix dynamics have different roles in driving budding in different organs. This historical review article summarizes the first founding literature data concerning branching morphogenesis occurring in kidney, lung, vascular system, mammary glands and neurons.