This overview gravitates on research achievements that have recently emerged from the confluence between Big Data technologies and bio-inspired computation. A manifold of reasons can be identified for the profitable synergy between these two paradigms, all rooted on the adaptability, intelligence and robustness that biologically inspired principles can provide to technologies aimed to manage, retrieve, fuse and process Big Data efficiently. We delve into this research field by first analyzing in depth the existing literature, with a focus on advances reported in the last few years. This prior literature analysis is complemented by an identification of the new trends and open challenges in Big Data that remain unsolved to date, and that can be effectively addressed by bio-inspired algorithms. As a second contribution, this work elaborates on how bio-inspired algorithms need to be adapted for their use in a Big Data context, in which data fusion becomes crucial as a previous step to allow processing and mining several and potentially heterogeneous data sources. This analysis allows exploring and comparing the scope and efficiency of existing approaches across different problems and domains, with the purpose of identifying new potential applications and research niches. Finally, this survey highlights open issues that remain unsolved to date in this research avenue, alongside a prescription of recommendations for future research.