This invention relates to novel three-dimensional (3D) carbon fibers which are original (or primary) carbon fibers (OCF) with secondary carbon filaments (SCF) grown thereon, and, if desired, tertiary carbon filaments (TCF) are grown from the surface of SCF forming a filamentous carbon network with high surface area. The methods and apparatus are provided for growing SCF on the OCF by thermal decomposition of carbonaceous gases (CG) over the hot surface of the OCF without use of metal-based catalysts. The thickness and length of SCF can be controlled by varying operational conditions of the process, e.g., the nature of CG, temperature, residence time, etc. The optional activation step enables one to produce 3D activated carbon fibers with high surface area. The method and apparatus are provided for growing TCF on the SCF by thermal decomposition of carbonaceous gases over the hot surface of the SCF using metal catalyst particles.