The polar sea ice packs play a key role in earth's ocean-climate system, and are sensitive indicators of climatic change. As a material, sea ice is a porous composite of pure ice with brine and air inclusions, whose microstructural properties depend strongly on temperature. The transport of brine, which carries salt, heat, and nutrients through sea ice, controls a broad range of geophysical, oceanographic, and biological processes. However, measurements of the fluid permeability of sea ice are sparse and little is known theoretically. We give mathematical formulations of the two key problems of fluid transport in sea ice: bulk flow of brine, and diffusion of dissolved substances such as pollutants or bacterial enzymes. We present a comprehensive theory for the fluid permeability of sea ice, based on rigorous bounds, continuum percolation theory, hierarchical models, network simulation, and microstructural imaging. Our theoretical results closely capture laboratory and Arctic field data. The role of fluid advection in thermal transport through sea ice is also discussed. This work is joint with Hajo Eicken, Geophysical Institute, University of Alaska Fairbanks, Jingyi Zhu, Department of Mathematics, University of Utah, and four University of Utah undergraduate students in the Mathematics REU program.