It is challenging to obtain high-quality dispersions of single-wall nanotubes (SWNTs) in composite matrix materials, in order to reach the full potential of mechanical and electronic properties. The most widely used matrix materials are polymers, and the route to achieving high quality dispersions of SWNT is mainly chemical functionalization of the SWNT. This leads to increased cost, a loss of strength and lower conductivity. In addition full potential of colloidal self-assembly cannot be fully exploited in a polymer matrix. This may limit the possibilities for assembly of highly ordered structural nanocomposites. Here we show that nanofibrillated cellulose (NFC) can act as an excellent aqueous dispersion agent for as-prepared SWNTs, making possible low-cost exfoliation and purification of SWNTs with dispersion limits exceeding 40 wt %. The NFC:SWNT dispersion may also offer a cheap and sustainable alternative for molecular self-assembly of advanced composites. We demonstrate semitransparent conductive films, aerogels and anisotropic microscale fibers with nanoscale composite structure. The NFC:SWNT nanopaper shows increased strength at 3 wt % SWNT, reaching a modulus of 13.3 GPa, and a strength of 307 MPa. The anisotropic microfiber composites have maximum conductivities above 200 S cm-1 and current densities reaching 1400 A cm-2