It is known that the group of projectivities G of a B-oval B=(M, F) is triply-transitive on the set M. Using the classification of all finite triply-transitive groups we list the possible groups of projectivities of the finite B-ovals. Furthermore, we give a definition of hyperbolic parts that covers hyperbolic parts derived from finite B-ovals of odd order. We also list the possible groups of projectivities of finite hyperbolic parts. Following this we define algebraic B-ovals and Lie B-ovals and show that both classes contain only B-conics. Finally, we investigate real B-ovals.