  The aim of this paper is to generalize the theory of nonlinear connections of
Grifone ([3] and [4]). We adopt the point of view of Anona [1] and continue
developing the approach established by the first author in [10].
  The first part of the work is devoted to the problem of associating to each
$L$-regular linear connection on $M$ a nonlinear $L$-connection on $M$. The
route we have followed is significantly different from that of Grifone. We
introduce an almost-complex and an almost-product structures on $M$ by means of
a given $L$-regular linear connection on $M$. The product of these two
structures defines a nonlinear $L$-connection on $M$, which generalizes
Grifone's nonlinear connection.
  The seconed part is devoted to the converse problem: associating to each
nonlinear $L$-connection \G on $M$ an $L$-regular linear connection on $M$;
called the $L$-lift of \G. The existence of this lift is established and the
fundamental tensors associated with it are studied.
  In the third part, we investigate the $L$-lift of a homogeneous
$L$-connection \G, called the Berwald $L$-lift of \G. Then we particularize our
study to the $L$-lift of a conservative $L$-connection. This $L$-lift enjoys
some interesting properties. We finally deduce various identities concerning
the curvature tensors of such a lift.
  Grifone's theory can be retrieved by letting $M$ be the tangent bundle of a
differentiable manifold and $L$ be the natural almost-tangent structure $J$ on
$M$.
