Network-based biocomputation (NBC) is an alternative, parallel computation approach that can potentially solve technologically important, combinatorial problems with much lower energy consumption than electronic processors. In NBC, a combinatorial problem is encoded into a physical, nanofabricated network. The problem is solved by biological agents (such as cytoskeletal filaments driven by molecular motors) that explore all possible pathways through the network in a massively parallel and highly energy-efficient manner. Whereas there is currently a rapid development in the size and types of problems that can be solved by NBC in proof-of-principle experiments, significant challenges still need to be overcome before NBC can be scaled up to fill a technological niche and reach an industrial level of manufacturing. Here, we provide a roadmap that identifies key scientific and technological needs. Specifically, we identify technology benchmarks that need to be reached or overcome, as well as possible solutions for how to achieve this. These include methods for large-scale production of nanoscale physical networks, for dynamically changing pathways in these networks, for encoding information onto biological agents, for single-molecule readout technology, as well as the integration of each of these approaches in large-scale production. We also introduce figures of merit that help analyze the scalability of various types of NBC networks and we use these to evaluate scenarios for major technological impact of NBC. A major milestone for NBC will be to increase parallelization to a point where the technology is able to outperform the current run time of electronic processors. If this can be achieved, NBC would offer a drastic advantage in terms of orders of magnitude lower energy consumption. In addition, the fundamentally different architecture of NBC compared to conventional electronic computers may make it more advantageous to use NBC to solve certain types of problems and instances that are easy to parallelize. To achieve these objectives, the purpose of this roadmap is to identify pre-competitive research domains, enabling cooperation between industry, institutes, and universities for sharing research and development efforts and reducing development cost and time.