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This thesis aims to explore the properties of the motion of finite size,compact test bodies around a Kerr black hole in the small mass-ratioapproximation. The small body is modelled as a perturbation of Kerr geometry,neglecting its gravitational back-reaction but including deviations from apurely geodesic motion by allowing it to possess a non-trivial internalstructure. Such a body can be accurately described by a worldline endowed witha collection of multipole moments. Hereafter, we shall always consider themultipole expansion truncated at quadrupole order. Moreover, only spin-inducedquadrupole moment will be taken into account, thus discarding the presence ofany tidal-type deformation. For astrophysically realistic objects, thisapproximation is consistent with expanding the equations of motion up to secondorder in the body's spin magnitude.  The text is structured as follows. The first part is devoted to an extendedreview of geodesic motion in Kerr spacetime, including Hamiltonian formulationand classification of timelike geodesics, with a particular emphasis put onnear-horizon geodesics of high spin black holes. The second part introduces theequations of motion for extended test bodies in generic curved spacetime, alsoknown as Mathisson-Papapetrou-Dixon (MPD) equations. The third part discussesconserved quantities for the MPD equations in Kerr spacetime, restricting tothe aforementioned quadrupole approximation. Finally, the covariant Hamiltonianformulation of test body motion in curved spacetime is presented, and anHamiltonian reproducing the spin-induced quadrupole MPD equations is derived.It is shown that the constants of motion obtained in the previous part directlyarise while solving the Hamilton-Jacobi equation at first order in the spinmagnitude. Some expectations regarding the computation at quadratic order closethe discussion.

The Motion of Test Bodies around Kerr Black Holes