  In this paper, we introduce a notion of a left-symmetric algebroid, which is
a generalization of a left-symmetric algebra from a vector space to a vector
bundle. The left multiplication gives rise to a representation of the
corresponding sub-adjacent Lie algebroid. We construct left-symmetric
algebroids from $\mathcal O$-operators on Lie algebroids. We study phase spaces
of Lie algebroids in terms of left-symmetric algebroids. Representations of
left-symmetric algebroids are studied in detail. At last, we study deformations
of left-symmetric algebroids, which could be controlled by the second
cohomology class in the deformation cohomology.
