This review critically introduces recent results on nanocomposite materials derived from the fibrous clay silicates sepiolite and palygorskite and combined with diverse types of polymers, from typical thermoplastics to biopolymers such as polysaccharides, proteins, lipids, and nucleic acids. First, the main features of both silicates are described, emphasizing the structural and textural characteristics that determine the interaction mechanisms with organic compounds and particularly with polymers, which define the final properties of the resulting materials. The crucial role of the clay–silicate interface governing the terminal properties of the nanocomposites is especially considered. Second, this work reports and discusses different experimental approaches and preparative procedures adopted for the nanofabrication and conformation (powders, films, monoliths, foams, etc.) of nanocomposites, comparing in certain cases with analogous materials derived from layered clays instead of sepiolite or palygorskite. Selected examples of fibrous clay-based nanocomposites are discussed to show the broad versatility of these materials in application fields as diverse as structural materials, conducting nanocomposites, biomaterials, environmental remediation, and sensor devices.