  We study the spreading of correlations and other physical quantities in
quantum lattice models with interactions or hopping decaying like $r^{-\alpha}$
with the distance $r$. Our focus is on exponents $\alpha$ between 0 and 6,
where the interplay of long- and short-range features gives rise to a complex
phenomenology and interesting physical effects, and which is also the relevant
range for experimental realizations with cold atoms, ions, or molecules. We
present analytical and numerical results, providing a comprehensive picture of
spatio-temporal propagation. Lieb-Robinson-type bounds are extended to strongly
long-range interactions where $\alpha$ is smaller than the lattice dimension,
and we report particularly sharp bounds that are capable of reproducing regimes
with soundcone as well as supersonic dynamics. Complementary lower bounds prove
that faster-than-soundcone propagation occurs for $\alpha<2$ in any spatial
dimension, although cone-like features are shown to also occur in that regime.
Our results provide guidance for optimizing experimental efforts to harness
long-range interactions in a variety of quantum information and signaling
tasks.
