  We pursue the current developments in random tensor theory by laying the
foundations of a free probability theory for tensors and establish its
relevance in the study of random tensors of high dimension. We give a
definition of freeness associated to a collection of tensors of possibly
different orders. Our definition reduces to the usual freeness when only
tensors of order 2 are concerned. We define the free cumulants which are
associated to this notion of tensor freeness. We prove that the basic models of
random tensors are asymptotically free as the dimension goes to infinity. On
the way, we establish Schwinger-Dyson loop equations associated to random
tensors.
