Eukaryotic cells possess numerous dynamic membrane-less organelles, RNP granules, enriched in RNA and RNA-binding proteins containing disordered regions. We demonstrate that the disordered regions of key RNP granule components and the full-length granule protein hnRNPA1 can phase separate in vitro, producing dynamic liquid droplets. Phase separation is promoted by low salt concentrations or RNA. Over time, the droplets mature to more stable states, as assessed by slowed fluorescence recovery after photobleaching and resistance to salt. Maturation often coincides with formation of fibrous structures. Different disordered domains can co-assemble into phase-separated droplets. These biophysical properties demonstrate a plausible mechanism by which interactions between disordered regions, coupled with RNA binding, could contribute to RNP granule assembly in vivo through promoting phase separation. Progression from dynamic liquids to stable fibers may be regulated to produce cellular structures with diverse physiochemical properties and functions. Misregulation could contribute to diseases involving aberrant RNA granules.