We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on ${\mathbb Z}^n$, where the jump rates are asymmetric and long-range of order $\|x\|^{-(n+α)}$ for a particle displacement of order $\|x\|$. Two types of evolution equations are identified depending on the strength of the long-range asymmetry. When $0<α<1$, we find a new integro-partial differential hydrodynamic equation, in an anomalous space-time scale. On the other hand, when $α\geq 1$, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.