Carbon nanotubes are potential building blocks for nanofluidic devices. The way fluids flow into and fill nanotubes is therefore of wide interest; however, there are currently no experimental data or theoretical models for this nanoscale process. We have carried out molecular dynamics simulations of nanotubes imbibing oil at an oil/vapor interface at 298 K. We find that nanotubes imbibe very rapidly (<or=445 m/s) via complex imbibition dynamics. The filling does not obey the macroscopic Washburn equation; the penetration length is a linear function of time.