EXCITABLE media are exemplified by a range of living systems(1-8), such as mammalian heart muscle(6) and its cells(1) and Xenopus eggs(2,3). They also occur in non-living systems such as the autocatalytic Belousov-Zhabatinsky reactiong(9-14). In most of these systems, activity patterns, such as concentration waves, typically radiate as spiral waves from a vortex of excitation created by some nonuniform stimulus. In three-dimensional systems, the vortex is commonly a line, and these vortex lines can form linked and knotted rings which contract into compact, particle-like bundles(9-30). In most previous work these stable ’organizing centres’ have been found to be symmetrical and can be classified topologically. Here I show through numerical studies of a generic excitable medium that the more general configuration of vortex lines is a turbulent tangle, which is robust against changes in the parameters of the system or perturbations to it. In view of their stability, I suggest that these turbulent tangles should be observable in any of the many known excitable media.