If a three-dimensional physical electronic system emulating synapse networks could be built, that would be a significant step toward neuromorphic computing. However, the fabrication complexity of complementary metal-oxide-semiconductor architectures impedes the achievement of three-dimensional interconnectivity, high-device density, or flexibility. Here we report flexible three-dimensional artificial chemical synapse networks, in which two-terminal memristive devices, namely, electronic synapses (e-synapses), are connected by vertically stacking crossbar electrodes. The e-synapses resemble the key features of biological synapses: unilateral connection, long-term potentiation/depression, a spike-timing-dependent plasticity learning rule, paired-pulse facilitation, and ultralow-power consumption. The three-dimensional artificial synapse networks enable a direct emulation of correlated learning and trainable memory capability with strong tolerances to input faults and variations, which shows the feasibility of using them in futuristic electronic devices and can provide a physical platform for the realization of smart memories and machine learning and for operation of the complex algorithms involving hierarchical neural networks.