In this paper, we prove a family of Lieb-Robinson bounds for discrete spin systems with long-range interactions. Our results apply for arbitrary -body interactions, so long as they decay with a power law greater than , where is the dimension of the system. More precisely, we require that the sum of the norm of terms with diameter greater than or equal to , acting on any one site, decays as a power law , with . These bounds allow us to prove that, at any fixed time, the spatial decay of a time evolved operator follows arbitrarily closely to . Moreover, we introduce an alternative light cone definition for power-law interacting quantum systems which captures the region of the system where changing the Hamiltonian can affect the evolution of a local operator. In short-range interacting systems, this light cone agrees with the conventional definition. However, in long-range interacting systems, our definition yields a stricter light cone, which is more relevant in most physical contexts.