In recent years, Artificial Intelligence has shifted towards edge computing paradigm, where systems compute data in real-time on the edge of the network, close to the sensor that acquires them. The requirements of a system operating on the edge are very tight: power efficiency, low area footprint, fast response times, and online learning. Moreover, in order to fully optimize sensor performance and broaden applications by developing smart wearable and implantable devices, solutions must be compatible with flexible substrates. Brain-inspired architectures such as Spiking Neural Networks (SNNs) use artificial neurons and synapses that perform low-latency computation and internal-state storage simultaneously with very low power consumption. However, SNNs at present are mainly implemented on standard CMOS technologies, which makes it challenging to meet the above-mentioned constraints. In this respect, memristive technology has shown promising results, due to its ability to support fast and energy-efficient non-volatile storage of the SNN parameters in a nanoscale footprint. In this perspective work, the main challenges to achieve a neuromorphic-memristive hardware are presented, particularly in the context of optimizing such systems for applications on the edge. The aspects to be considered for integration with flexible substrates will also be discussed.