  We introduce the notion of a nonlinear splitting on a fibre bundle as a
generalization of an Ehresmann connection. We present its basic properties and
we pay attention to the special cases of affine, homogeneous and principal
nonlinear splittings. We explain where nonlinear splittings appear in the
context of Lagrangian systems and Finsler geometry and we show their relation
to Routh symmetry reduction, submersive second-order differential equations and
unreduction. We define a curvature map for a nonlinear splitting, and we
indicate where this concept appears in the context of nonholonomic systems with
affine constraints and Lagrangian systems of magnetic type.
