The side walls of filaments of Chaetomorpha melagonium, Ch. princeps, Cladophora prolifera and Cl. rupestris are examined to detect changes in the crossed-microfibrillar structure associated with cell extension, using X-ray diffraction analysis together with light and electron microscopy. It is demonstrated that the outermost wall layers may be cracked or even torn apart so that the wall must be stretched during cell growth. In passing from inner through central to outer wall layers of the cylindrical cells, the cellulose microfibrils running in a slow spiral become more and more disoriented and the spiral becomes steeper; those running in a steep spiral become more nearly parallel to each other and the spiral becomes flatter. These changes are clearly reactions to cell elongation. In outer layers, the microfibrils in slow spirals are arranged in more perfect uniplanar orientation than they are in inner layers. These microfibrils must therefore turn about their lengths as the lamellae, of which they are a part, extend. The flattening of the steeper spirals suggests that the filaments twist, as they elongate, in a growth spiral of the same sign as that of the steeper structural spiral. Such spiral growth has been verified in all four species. The rate both of length increase ($\Delta $L/$\Delta $t) of a filament and of the rotation of one end with respect to the other ($\Delta \phi $/$\Delta $t) varies with varying external factors but the ratio $\Delta \phi $/$\Delta $L remains approximately constant for any one filament. $\Delta \phi $/$\Delta $L varies between filaments approximately inversely to cell diameter. Replacement of sea water by hypertonic solutions causes the rotation to be reversed, and by hypotonic solutions to proceed in the normal sense at a higher rate. It is shown, under certain simplifying assumptions, that the changes in spiral organization through the wall and the rates of spiral growth are quantitatively related. If rotation of one end of a filament with respect to the other is prevented during growth, changes in the spiral organization then become erratic. All these observations recall earlier work on sporangiophores of Phycomyces and confirm that spiral growth is a consequence of anistropic stresses induced in the walls by turgor pressure in the cells. In these firm walls, the microfibrils undergo movements of translation and rotation as a consequence of cell extension, in general harmony the multi-net growth hypothesis.