Biological and artificial neural networks are meant to benefit from high resilience to noise, and to display great performance when learning tasks are considered. The advantage of a hardware-based artificial neural network, against its software counterpart, is that it is able to achieve similar results with a significant reduction of size, time and power consumption. In this work, the first RRAM-based self-organizing and topographic neuromorphic system is proposed. An automatic characterization setup has been developed for the study of our RRAM devices response to pulse-programming. An extension to a static compact model for non-linear memristive devices is provided. This extension allows including variability effects and has been used for performing crossbar arrays simulations. Inspiration in the biological mechanisms involved in sensory processing is taken for adapting the self-organizing map algorithm, commonly used in artificial intelligence, to achieve topographical organization. Results support that our RRAM-based neuromorphic system has significant tolerance to device variability.