High yields of graphitic conical nanofibres (5-70 nm OD; <5 mum long) are produced by pyrolysing palladium precursors under Ar at 850-1000 degreesC. The fibres exhibit diamond-shaped Pd particles at their tips, which are responsible for the formation of stacked graphene cones (open, lampshade-type, or closed). The cones observed with apex angles of ca. 30 degrees, 50 degrees and 70 degrees can be explained by an open cone approach, which considers different chiralities. Due to the presence of open edges (dangling bonds), we envisage that these novel nanofibres may find important applications in the fabrication of field emitters, gas storage components and composites.