This chapter describes, in some detail, properties of electrically neutral and electrically charged, rotating black holes predicted in Einstein’s theory of gravitation and the nature of their geodesics. In Sect. 31.1 we introduce the reader to a 2-parameter family of axially symmetric Kerr spacetime metric tensor fields in terms of a chart used by Kerr in his 1963 paper. After a discussion of oblate spheroidal coordinates for Euclidean \(\mathbb {R}^{3}\) , the Boyer-Lindquist chart for the Kerr metric is used to calculate Kretschmann scalars that expose a geometric singularity. It is shown that this is a “ring singularity” on a spacetime hypersurface in a unique equatorial plane. It is argued that certain spacetime domains are stationary relative to certain observers (see Sect. 30.11 ), but others are non-stationary. Section 31.2 is devoted to a detailed analysis of Kerr spacetimes in the Kerr-Boyer-Lindquist charts and a calculation of the Komar periods associated with particular Killing vector fields is followed by a definition of the inner and outer Kerr ergospheres, and the inner and outer Kerr event horizons, leading to the notion of a null, stationary, axially symmetric hypersurface or Killing horizon. A computation of the “area” of a 2-sphere in this hypersurface leads to an identity, known as the Smarr formula. The computations are then exploited in Sect. 31.3 where we construct a Killing tensor field and a first-order ODE system for causal Kerr spacetime geodesics. In Sect. 31.4 an analysis of timelike and null equatorial Kerr geodesics is given, culminating in a single second-order integrable “relativistic Binet type” ODE for all equatorial orbits given suitable initial conditions. Particular numerical solutions to these planar orbits are displayed for specific values of the Kerr metric parameters and all constants of the motion. Particular attention is drawn to a phenomenon called “frame dragging” that has been attributed to “rotating Kerr space” since it is not evident in the behaviour of equatorial Schwarzschild geodesics. A class of timelike Kerr geodesics that describes particle orbits outside the event horizon but do not lie in the unique Kerr equatorial plane are discussed in Sect. 31.5. A particular example is shown that illustrates why such curves are called “vortical geodesics”. In Sect. 31.6, an exterior Kerr-Newman solution is verified in a Boyer-Lindquist chart and shown to describe an exact axially symmetric, stationary analytic solution of the Einstein-Maxwell tensor system for the Maxwell 2-form F and metric tensor \(\mathtt {g}\) . It is conjectured that such solutions could lead to the existence of “electrically charged black holes”.