In an earlier paper, Blennerhassett & Hall (1979) investigated the linear stability of the flow between concentric cylinders of finite length. The inner cylinder was taken to rotate, while the outer cylinder was fixed. Furthermore, the end walls rotated such that the flow was purely circumferential. In this paper the finite amplitude development of the unstable disturbances to the flow is considered. It is found that the usual perturbation expansion of nonlinear stability theory must be modified if the cylinders are not infinitely long. The bifurcation to a Taylor vortex flow in finite cylinders is shown to be two-sided. The latter effect is shown to be a direct consequence of the finiteness of the cylinders and by taking the cylinders to be very long, we recover the results obtained previously for the infinite problem. The interaction of the two most dangerous modes of linear theory is also investigated. For certain values of the length of the cylinders the initial finite amplitude Taylor vortex flow is shown to become unstable to another class of axisymmetric disturbances. The effect of perturbing the end conditions towards the no-slip conditions appropriate to most experimental configurations is also discussed. Some discussion of the instability problem in very long cylinders with fixed ends is given.