We review the present state of polymer nanocomposites research in which the fillers are single-wall or multiwall carbon nanotubes. By way of background we provide a brief synopsis about carbon nanotube materials and their suspensions. We summarize and critique various nanotube/polymer composite fabrication methods including solution mixing, melt mixing, and in situ polymerization with a particular emphasis on evaluating the dispersion state of the nanotubes. We discuss mechanical, electrical, rheological, thermal, and flammability properties separately and how these physical properties depend on the size, aspect ratio, loading, dispersion state, and alignment of nanotubes within polymer nanocomposites. Finally, we summarize the current challenges to and opportunities for efficiently translating the extraordinary properties of carbon nanotubes to polymer matrices in hopes of facilitating progress in this emerging area.