The brain can hold the eyes still because it stores a memory of eye position. The brain's memory of horizontal eye position appears to be represented by persistent neural activity in a network known as the neural integrator, which is localized in the brainstem and cerebellum. Existing experimental data are reinterpreted as evidence for an ''attractor hypothesis'' that the persistent patterns of activity observed in this network form an attractive line of fixed points in its state space. Line attractor dynamics can be produced in linear or nonlinear neural networks by learning mechanisms that precisely tune positive feedback.The brain moves the eyes with quick saccadic movements. Between saccades, it holds the eyes still, a capability that depends on a ''memory'' of angular position of the eyes. A person who has lost the ability to store this memory can hold his or her eyes still at only a single null position. After saccades to other positions, the eyes drift back toward the null position, in a disorder known as gaze-evoked nystagmus (1). Localization studies suggest that the memory of eye position is stored in a neural network that extends over several areas in the brainstem and cerebellum (2-19).When the eyes are still, the pattern of neural activity in the memory network is constant in time. For every position of the eyes, the pattern of activity is different, and can persist much longer than the intrinsic persistence time of a single neuron's activity (20)(21)(22). Therefore, the long persistence of network activity appears to be a collective effect that depends on the interactions between the neurons of the memory network.Based on existing experimental data, it will be argued that the memory of eye position is stored in a neural network with an approximate line attractor dynamics. If synaptic strengths and other parameters are precisely tuned by learning mechanisms, a linear network (23-26) can exactly realize a line attractor dynamics, and a nonlinear network (27, 28) can achieve a good approximation.