The study of geodesics is of intrinsic significance in the study of the geometry of space-time. In this thesis null, space-like and time-like geodesics are studied in the case of the space-times of Schwarzschild, Reissner-Nordstrouml;m and Kerr black holes. These space-times have been investigated with varying degrees of thoroughness in many articles and some books. However, there are some significant gaps in these treatments and the central aim of this thesis is to fill these gaps where necessary. Moreover, the following topics are covered for the first time. 1. In Chapter 4 a thorough treatment of the space-like geodesics of the Schwarzschild solutions has been given. These geodesics are the trajectories of Tachyons (faster than light particles) and are treated in a complete manner. This has been done by obtaining exact solutions and solving them numerically. 2. In Part II all solutions for geodesics for a Reissner-Nordstrouml;m black hole have been given in complete detail, i.e. time-like, null and space-like geodesics and orbit of a charged particle. 3. In Chapter 14 all solutions for geodesics in the equatorial plane of a Kerr black hole have been given in complete detail, i.e. time-like, null and space-like geodesics. 4. The study of special types of non-equatorial geodesics for a Kerr black hole have been given in complete detail, i.e. time-like (Chapter 17), null (Chapter 15) and space-like (Chapter 16). This has been done in order to distinguish the qualitatively different types of solutions. Calculation of the explicit formulas, which describe these geodesics, as well as numerically computed diagrams representing the geodesics have been incorporated in these studies. The following subjects have been also treated: 5. Solutions for the geodesics in Reissner-Nordstrouml;m black holes with |Q_*| gt;= M, which are black holes with one (|Q_*| = M) or no horizon (|Q_*|gt; M) (Chapter 8). 6. Solutions of geodesics in extreme and fast Kerr black holes, i.e. black holes with a = M (extreme) and a gt; M (fast). As in the case of |Q_*| gt; M, fast black holes have naked singularities (Chapter 14). 7. Some general observations about orbit types of the Kerr black holes regarding relationships between parameters such as angular momentum, energy, Carter constant and mass and angular momentum of black holes (Chapter 13). 8. Some corrections to errors found in the literature. While it has not been possible to cover all different cases which occur for possible relations amongst the parameters specifying a general black hole, interesting geodesics have, however, been studied and a more thorough presentation of the properties of geodesics has now been given.