Devices and methods for removing cerebrospinal fluid (CSF) from a CSF space of a patient at relatively constant flow rates for patients having normal intracranial pressures, e.g. patients not suffering from hydrocephalus. The devices and methods provide drainage paths which permit the removal of CSF at relatively low flow rates, usually below 0.2 ml/min, at normal intracranial pressures, e.g. an intracranial pressure between -170 mm of H.sub.2 O in upright patients and 200 mm of H.sub.2 O in reclining patients. The removal of CSF at relatively low, constant rates is particularly suitable for treating Alzheimer's disease and other conditions related to the presence of toxic and/or pathogenic substances in the CSF.