Current electrophysiological or optical techniques cannot reliably perform simultaneous intracellular recordings from more than a few tens of neurons. Here we report a nanoelectrode array that can simultaneously obtain intracellular recordings from thousands of connected mammalian neurons in vitro. The array consists of 4,096 platinum-black electrodes with nanoscale roughness fabricated on top of a silicon chip that monolithically integrates 4,096 microscale amplifiers, configurable into pseudocurrent-clamp mode (for concurrent current injection and voltage recording) or into pseudovoltage-clamp mode (for concurrent voltage application and current recording). We used the array in pseudovoltage-clamp mode to measure the effects of drugs on ion-channel currents. In pseudocurrent-clamp mode, the array intracellularly recorded action potentials and postsynaptic potentials from thousands of neurons. In addition, we mapped over 300 excitatory and inhibitory synaptic connections from more than 1,700 neurons that were intracellularly recorded for 19 min. This high-throughput intracellular-recording technology could benefit functional connectome mapping, electrophysiological screening and other functional interrogations of neuronal networks.