In a present context of sustainable energy and hazard mitigation, understanding fluid migration in sedimentary basins-large subsea provinces of fine saturated sands and clays-is a crucial challenge. Such migration leads to gas or liquid expulsion at the seafloor, which may be the signature of deep hydrocarbon reservoirs, or precursors for violent subsea fluid releases. If the former may orient future exploitation, the latter represent strong hazards for anthropic activities such as offshore production, CO2 storage, transoceanic telecom fibers or deep-sea mining. However, at present, the dynamics of fluid migration in sedimentary layers, in particular the upper 500 m, still remains unknown in spite of its strong influence on the fluid distribution at the seafloor. Understanding the mechanisms controlling the fluid migration and release needs the combination of accurate field data, laboratory experiments and numerical simulations. Each technique shall lead to the understanding of the fluid structures, the mechanisms at stake, and a deep insight on the fundamental processes ranging from the grain scale to the kilometers-long natural pipes in the sedimentary layers. Here we review the present available techniques, advances and challenges still open for the geosciences, physics, and computer science communities.