Brain-inspired neuromorphic computing systems fueled the emergence of memristor-based artificial synapses, however, conventional silicon-based devices restricted their usage in the wearable field because of their difficulty in bending. To tackle the above challenge, a vertically structured flexible memristor with aluminum-based hydroquinone organic-inorganic hybrid film and Al2O3 as the functional layer, ITO and Pt as the bottom and top electrodes, and PET as the substrate has been developed utilizing molecular/atomic layer deposition to achieve a tradeoff between the resistive transition properties and the flexibility of memristors. The obtained devices combine stable resistive switching behavior and flexibility, showing high switching ratio of 103, better retention (up to 105 s) and endurance properties (up to 104 cycles), and robustness at radius of curvature of 4.5 mm after 104 bending cycles. Furthermore, the presence of multilevel resistive states in these devices ensures that the memristor can emulate synaptic properties such as paired-pulse facilitation, transition from short-term plasticity to long-term plasticity, long-term potentiation and depression, and spike-time-dependent plasticity. The resistive switching mechanism and the role of the bending state on the electrical performance of the device are explored. The fully connected artificial neural network based on the memristor can achieve a recognition accuracy of 90.2% for handwritten digits after training and learning. Flexible memristor will bring feasible advances to the integration of neuromorphic computing and wearable functionality.