Capillary phenomena occur in both natural and human-made systems, from equilibria in the presence of solids (grains, walls, metal wires) to multiphase flows in heterogeneous and fractured porous media. This book, composed of two volumes, develops fluid mechanics approaches for two immiscible fluids (water/air or water/oil) in the presence of solids (tubes, joints, grains, porous media). Their hydrodynamics are typically dominated by capillarity and viscous dissipation. This first volume presents the basic concepts and investigates two-phase equilibria, before analyzing two-phase hydrodynamics in discrete and/or statistical systems (tubular pores, planar joints). It then studies flows in heterogeneous and stratified porous media, such as soils and rocks, based on Darcy's law. This analysis includes unsaturated flow (Richards equation) and two-phase flow (Muskat equations). Overall, the two volumes contain basic physical concepts, theoretical analyses, field investigations and statistical and numerical approaches to capillary-driven equilibria and flows in heterogeneous systems.