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We discuss a particular higher order gravity theory, Lovelock theory, thatgeneralises in higher dimensions, general relativity. After briefly motivatingmodifications of gravity, we will introduce the theory in question and we willargue that it is a unique, mathematically sensible, and physically interestingextension of general relativity. We will see, by using the formalism ofdifferential forms, the relation of Lovelock gravity to differential geometryand topology of even dimensional manifolds. We will then discuss a genericstaticity theorem, which will give us the charged static black hole solutions.We will examine their asymptotic behavior, analyse their horizon structure andbriefly their thermodynamics. We will then examine the distributional matchingconditions for Lovelock theory. We will see how induced 4 dimensionalEinstein-Hilbert terms result on the brane geometry from the higher orderLovelock terms. With the junction conditions at hand, we will go back to theblack hole solutions and give applications for braneworlds: perturbations ofcodimension 1 braneworlds and the exact solution for braneworld cosmology aswell as the determination of maximally symmetric codimension 2 braneworlds. Inboth cases, the staticity theorem evoked beforehand will give us the generalsolution for braneworld cosmology in codimension 1 and maximal symmetry warpedbranes of codimension 2. We will then end with a discussion of the simplestKaluza-Klein reduction of Lovelock theory to a 4 dimensionalvector-scalar-tensor theory which has the unique property of retaining secondorder field equations. We will conclude by listing some open problems andcommon difficulties.

Higher order gravity theories and their black hole solutions