In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs...

We propose a definition of Leibniz cohomology, $H{L}^{*}$, for differentiable manifolds. Then $H{L}^{*}$ becomes a non-commutative version of Gelfand-Fuks cohomology. The calculations of $H{L}^{*}({\mathbf{R}}^n;\mathbf{R})$ reduce to those of formal vector fields, and can be identified with certain invariants of foliations.

We give a criterion for Leibniz elements in a free diassociative algebra. In the diassociative case one can consider two versions of Lie commutators. We give criterions for elements of diassociative algebras to be Lie under these commutators. One of them corresponds to Leibniz elements. It generalizes the Dynkin-Specht-Wever criterion for Lie elements in a free associative algebra.