  We discuss a particular higher order gravity theory, Lovelock theory, that
generalises in higher dimensions, general relativity. After briefly motivating
modifications of gravity, we will introduce the theory in question and we will
argue that it is a unique, mathematically sensible, and physically interesting
extension of general relativity. We will see, by using the formalism of
differential forms, the relation of Lovelock gravity to differential geometry
and topology of even dimensional manifolds. We will then discuss a generic
staticity theorem, which will give us the charged static black hole solutions.
We will examine their asymptotic behavior, analyse their horizon structure and
briefly their thermodynamics. We will then examine the distributional matching
conditions for Lovelock theory. We will see how induced 4 dimensional
Einstein-Hilbert terms result on the brane geometry from the higher order
Lovelock terms. With the junction conditions at hand, we will go back to the
black hole solutions and give applications for braneworlds: perturbations of
codimension 1 braneworlds and the exact solution for braneworld cosmology as
well as the determination of maximally symmetric codimension 2 braneworlds. In
both cases, the staticity theorem evoked beforehand will give us the general
solution for braneworld cosmology in codimension 1 and maximal symmetry warped
branes of codimension 2. We will then end with a discussion of the simplest
Kaluza-Klein reduction of Lovelock theory to a 4 dimensional
vector-scalar-tensor theory which has the unique property of retaining second
order field equations. We will conclude by listing some open problems and
common difficulties.
