  In this paper, first we introduce the notion of an omni-representation of a
Leibniz algebra $\g$ on a vector space $V$ as a Leibniz algebra homomorphism
from $\g$ to the omni-Lie algebra $\gl(V)\oplus V$. Then we introduce the
omni-cohomology theory associated to omni-representations and establish the
relation between omni-cohomology groups and Loday-Pirashvili cohomology groups.
