  We introduce a general framework for detecting genuine multipartite
entanglement and non full-separability in multipartite quantum systems of
arbitrary dimensions based on correlation tensors. Regarding genuine
multipartite entanglement our conditions are comparable to previous approaches
in the case of qubits while they show particular strength in the relatively
unexplored case of higher dimensional systems. In the case of non-full
separability our conditions prove to be advantageous in situations where more
than two-body correlations are relevant, where most previous conditions turned
out to be weak. Moreover, they allow for the detection of fully bound entangled
states. Finally, we also discuss experimentally-friendly ways of implementing
our conditions, which are based on directly measurable quantities.
