This course covers physical, mathematical, and simulation aspects of single and two-phase flow and transport through porous media. Conservation equations for multiphase, multicomponent flow. Modeling of fluid mechanical instabilities such as viscous fingering and density-driven convection. Coupling fluid flow with chemical reactions and heat transfer. Numerical methods for elliptic equations: finite volume methods, two-point flux approximations. Numerical methods for hyperbolic equations: high-order explicit methods, implicit method. Applications in hydrology, geological CO2 sequestration, and induced seismicity, among others, will be demonstrated.