Intricate interactions between kinetochores and microtubules are essential for the proper distribution of chromosomes during mitosis. A crucial long-standing question is how vertebrate kinetochores generate chromosome motion while maintaining attachments to the dynamic plus ends of the multiple kinetochore MTs (kMTs) in a kinetochore fibre. Here, we demonstrate that individual kMTs in PtK(1) cells are attached to the kinetochore outer plate by several fibres that either embed the microtubule plus-end tips in a radial mesh, or extend out from the outer plate to bind microtubule walls. The extended fibres also interact with the walls of nearby microtubules that are not part of the kinetochore fibre. These structural data, in combination with other recent reports, support a network model of kMT attachment wherein the fibrous network in the unbound outer plate, including the Hec1-Ndc80 complex, dissociates and rearranges to form kMT attachments.