Neural electrical activity is fundamental to brain function, underlying a range of cognitive and behavioral processes, including movement, perception, decision-making, and consciousness. Abnormal patterns of neural signaling often indicate the presence of underlying brain diseases. The variability among individuals, the diverse array of clinical symptoms from various brain disorders, and the limited availability of diagnostic classifications, have posed significant barriers to formulating reliable model of neural signals for diverse application contexts. Here, we present BrainWave, the first foundation model for both invasive and non-invasive neural recordings, pretrained on more than 40,000 hours of electrical brain recordings (13.79 TB of data) from approximately 16,000 individuals. Our analysis show that BrainWave outperforms all other competing models and consistently achieves state-of-the-art performance in the diagnosis and identification of neurological disorders. We also demonstrate robust capabilities of BrainWave in enabling zero-shot transfer learning across varying recording conditions and brain diseases, as well as few-shot classification without fine-tuning, suggesting that BrainWave learns highly generalizable representations of neural signals. We hence believe that open-sourcing BrainWave will facilitate a wide range of clinical applications in medicine, paving the way for AI-driven approaches to investigate brain disorders and advance neuroscience research.