Recurrent neural networks (RNNs) have proven to be indispensable for processing sequential and temporal data, with extensive applications in language modelling, text generation, machine translation, and time-series forecasting. Despite their versatility, RNNs are frequently beset by significant training expenses and slow convergence times, which impinge upon their deployment in edge AI applications. Reservoir computing (RC), a specialized RNN variant, is attracting increased attention as a cost-effective alternative for processing temporal and sequential data at the edge. RC's distinctive advantage stems from its compatibility with emerging memristive hardware, which leverages the energy efficiency and reduced footprint of analog in-memory and in-sensor computing, offering a streamlined and energy-efficient solution. This review offers a comprehensive explanation of RC's underlying principles, fabrication processes, and surveys recent progress in nano-memristive device based RC systems from the viewpoints of in-memory and in-sensor RC function. It covers a spectrum of memristive device, from established oxide-based memristive device to cutting-edge material science developments, providing readers with a lucid understanding of RC's hardware implementation and fostering innovative designs for in-sensor reservoir computing systems. Lastly, we identify prevailing challenges and suggest viable solutions, paving the way for future advancements in in-sensor RC technology.