  A construction is proposed for linear connections on non-commutative
algebras. The construction relies on a generalisation of the Leibnitz rules of
commutative geometry and uses the bimodule structure of $\Omega^1$. A special
role is played by the extension to the framework of non-commutative geometry of
the permutation of two copies of $\Omega^1$. The construction of the linear
connection as well as the definition of torsion and curvature is first proposed
in the setting of the derivations based differential calculus of Dubois-
Violette and then a generalisation to the framework proposed by Connes as well
as other non-commutative differential calculi is suggested. The covariant
derivative obtained admits an extension to the tensor product of several copies
of $\Omega^1$. These constructions are illustrated with the example of the
algebra of $ n \times n$ matrices.
