The design of hydrogen storage materials is one of the principal challenges that must be met before the development of a hydrogen economy. While hydrogen has a large specific energy, its volumetric energy density is so low as to require development of materials that can store and release it when needed. While much of the research on hydrogen storage focuses on metal hydrides, these materials are currently limited by slow kinetics and energy inefficiency. Nanostructured materials with high surface areas are actively being developed as another option. These materials avoid some of the kinetic and thermodynamic drawbacks of metal hydrides and other reactive methods of storing hydrogen. In this work, progress towards hydrogen storage with nanoporous materials in general and porous organic polymers in particular is critically reviewed. Mechanisms of formation for crosslinked polymers, hypercrosslinked polymers, polymers of intrinsic microporosity, and covalent organic frameworks are discussed. Strategies for controlling hydrogen storage capacity and adsorption enthalpy via manipulation of surface area, pore size, and pore volume are discussed in detail.