  We discuss a very general theory of gravity, of which Lagrangian is an
arbitrary function of the curvature invariants, on the brane. In general, the
formulation of the junction conditions (except for Euler characteristics such
as Gauss-Bonnet term) leads to the powers of the delta function and requires
regularization. We suggest the way to avoid such a problem by imposing the
metric and its first derivative to be regular at the brane, the second
derivative to have a kink, the third derivative of the metric to have a step
function discontinuity, and no sooner as the fourth derivative of the metric to
give the delta function contribution to the field equations. Alternatively, we
discuss the reduction of the fourth-order gravity to the second order theory by
introducing extra scalar and tensor fields: the scalaron and the tensoron. In
order to obtain junction conditions we apply two methods: the application of
the Gauss-Codazzi formalism and the application of the generalized
Gibbons-Hawking boundary terms which are appended to the appropriate actions.
In the most general case we derive junction conditions without assuming the
continuity of the scalaron and the tensoron on the brane. The derived junction
conditions can serve studying the cosmological implications of the higher-order
brane gravity models.
