Plants exhibit phenotypical plasticity. Their general body plan is genetically determined, but plant architecture and branching patterns are variable and can be adjusted to the prevailing environmental conditions. The modular design of the plant facilitates such morphological adaptations. The prerequisite for the formation of a branch is the initiation of an axillary meristem. Here, we review the current knowledge about this process. After its establishment, the meristem can develop into a bud which can either become dormant or grow out and form a branch. Many endogenous factors, such as photoassimilate availability, and exogenous factors like nutrient availability or shading, have to be integrated in the decision whether a branch is formed. The underlying regulatory network is complex and involves phytohormones and transcription factors. The hormone auxin is derived from the shoot apex and inhibits bud outgrowth indirectly in a process termed apical dominance. Strigolactones appear to modulate apical dominance by modification of auxin fluxes. Furthermore, the transcription factor BRANCHED1 plays a central role. The exact interplay of all these factors still remains obscure and there are alternative models. We discuss recent findings in the field along with the major models.Plant architecture is economically significant because it affects important traits of crop and ornamental plants, as well as trees cultivated in forestry or on short rotation coppices. As a consequence, plant architecture has been modified during plant domestication. Research revealed that only few key genes have been the target of selection during plant domestication and in breeding programs. Here, we discuss such findings on the basis of various examples. Architectural ideotypes that provide advantages for crop plant management and yield are described. We also outline the potential of breeding and biotechnological approaches to further modify and improve plant architecture for economic needs.