The complete set of analytic solutions of the geodesic equation in a Schwarzschild-(anti-)de Sitter space-time is presented. The solutions are derived from the Jacobi inversion problem restricted to the set of zeros of the theta function, called the theta divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The different types of the resulting orbits are characterized in terms of the conserved energy and angular momentum as well as the cosmological constant. Using the analytical solution, the question whether the cosmological constant could be a cause of the Pioneer anomaly is addressed. The periastron shift and its post-Schwarzschild limit is derived. The developed method can also be applied to the geodesic equation in higher dimensional Schwarzschild space-times.

The complete analytical solutions of the geodesic equations in Kerr-de~Sitter and Kerr-anti-de~Sitter space-times are presented. They are expressed in terms of Weierstrass elliptic $\wp${}, $\zeta${}, and $\sigma${} functions as well as hyperelliptic Kleinian $\sigma${} functions restricted to the one-dimensional $\theta${} divisor. We analyze the dependency of timelike geodesics on the parameters of the space-time metric and the test-particle and compare the results with the situation in Kerr space-time with vanishing cosmological constant. Furthermore, we systematically can find all last stable spherical and circular orbits and derive the expressions of the deflection angle of flyby orbits, the orbital frequencies of bound orbits, the periastron shift, and the Lense-Thirring effect.

The complete analytical solutions of the geodesic equation of massive test particles in higher dimensional Schwarzschild, Schwarzschild–(anti)de Sitter, Reissner-Nordström and Reissner–Nordström–(anti)de Sitter spacetimes are presented. Using the Jacobi inversion problem restricted to the theta divisor the explicit solution is given in terms of Kleinian sigma functions. The derived orbits depend on the structure of the roots of the characteristic polynomials which depend on the particle’s energy and angular momentum, on the mass and the charge of the gravitational source, and the cosmological constant. We discuss the general structure of the orbits and show that due to the specific dimension-independent form of the angular momentum and the cosmological force a rich variety of orbits can emerge only in four and five dimensions. We present explicit analytical solutions for orbits up to 11 dimensions. A particular feature of Reissner-Nordström spacetimes is that bound and escape orbits traverse through different universes.

The motion of charged test particles in the gravitational field of a rotating and electromagnetically charged black hole as described by the Kerr-Newman metric is considered. We completely classify the colatitudinal and radial motion on the extended manifold $-$\infty${}$\le${}r$\le${}$\infty${}$, including orbits crossing the horizons or $r=0$. Analytical solutions of the equations of motion in terms of elliptic functions are presented which are valid for all types of orbits.

We investigate the motion of spinning test bodies in General Relativity. By means of a multipolar approximation method for extended test bodies we derive the equations of motion and classify the orbital motion of pole-dipole test bodies in the equatorial plane of the Kerr geometry. An exact expression for the periastron shift of a spinning test body is given. Implications of test body spin corrections are studied and compared with the results obtained by means of other approximation schemes.