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This course focuses on what happens when CO2 is injected into the subsurface to prevent its release to to the atmosphere. The mathematical theory describes subsurface flow of mixtures of a number of chemical components that form two phases. Applications of the theory cover many areas: carbon capture and geologic storage of CO2 in deep aquifers or in depleted oil or gas reservoirs, enhanced oil recovery by gas injection, contaminant transport in aquifers, and chromatography. Key topics include: Derivation of conservation equations in any coordinate system, and in dimensionless form; Convection and dispersion (physics of dispersion, CD equation and solution, measurement of dispersion coefficient, scaling of dispersion); Dispersion-free displacements (two phases, with two, three, four and more components, with component transfers between phases); Systems of first order pde's (eigenvalues are velocities at which compositions move, eigenvectors reveal allowable composition variations); Multicontact miscible displacement in enhanced oil recovery processes; Estimates of emission reductions associated with CO2 injection in aquifers and depleted oil and gas reservoirs.