The invention provides CVD-based methods for growing single-walled or multi-walled carbon nanotubes. In the methods of the invention, the nanotube growth environment is separated from the carbon-containing gas feed environment using a membrane which is substantially impermeable to gas flow but permits diffusion of carbon through the membrane. A catalyst for carbon nanotube growth is located on the growth side of the membrane while a catalyst for decomposition of carbon-containing gas is located on the feed side of the membrane. A path for diffusion of carbon through the membrane is provided between the growth and decomposition catalysts. Control of the size and shape of the carbon nanotube growth catalyst enables control over the nanotube structure formed.