ByoPiC is an ERC-funded project that aims at solving a key issue in astrophysics and cosmology: Where and how are the baryons hidden today? Answers to this questions will be provided by the detection, mapping, and assessment of the physical properties of warm and hot ionised baryons at large cosmic scales. It will be based on a statistically consistent, joint analysis of complementary multiwavelength data: Planck observations tracing hot, ionised baryons via the Sunyaev-Zeldovich effect, optimally combined with optical and near infrared galaxy surveys as tracers of cold baryons. A crucial key to the characterisation of baryonic structure on the largest cosmological scales is the development of innovative statistical tools, indispensable for the recovery of all the (cross-)information contained in these data. These tools will be used for the joint analysis of the data and will enable to detect cosmic web elements such as (super)clusters and filaments and assess their baryon content. The newly detected elements will be assembled to reconstruct the cosmic web as traced by both hot ionised baryons and galaxies. The most complete and detailed assessment of the census and contribution of warm-hot ionised baryons to the total baryon budget will be achieved. At the same time, the main physical processes driving the evolution of baryons in the cosmic web will be made possible.