Cell wall composition and architecture are different, with the former comprising the inventory of building blocks and their structures while the latter is a description of the higher-order structures that arise when the building blocks are put together and make a cell wall. Assembly of the wall components takes place, to a large extent, outside the plasma membrane and thus depends on self-assembly properties of the building blocks. Studies of cell wall architecture is a classical field in plant biology - this chapter covers ca 150 years. A core ambition of the studies was always to propose an architecture, a cell wall model, that was compatible with wall expansion as it was observed in elongating cells. Discovery of the plant hormone auxin, which could be used to stimulate wall expansion, made this hormone a key tool in cell wall research and auxin physiology and the quest for a cell wall model became intertwined fields. Four primary wall models have been proposed over the past 50 years and a discussion of these forms the core of the chapter. Even the newest model has little to offer in terms of helping us understand wall assembly, and we need to have a model in mind to pit ideas against, so the era of model building is not over.