  As with entanglement, different forms of Bell nonlocality arise in the
multipartite scenario. These can be defined in terms of relaxations of the
causal assumptions in local hidden-variable theories. However, a
characterisation of all the forms of multipartite nonlocality has until now
been out of reach, mainly due to the complexity of generic multipartite causal
models. Here, we employ the formalism of Bayesian networks to reveal
connections among different causal structures that make a both practical and
physically meaningful classification possible. Our framework holds for
arbitrarily many parties. We apply it to study the tripartite scenario in
detail, where we fully characterize all the nonlocality classes. Remarkably, we
identify new highly nonlocal causal structures that cannot reproduce all
quantum correlations. This shows, to our knowledge, the strongest form of
quantum multipartite nonlocality known to date. Finally, as a by-product
result, we derive a non-trivial Bell-type inequality with no quantum violation.
Our findings constitute a significant step forward in the understanding of
multipartite Bell nonlocality and open several venues for future research.
