Graphene, a two-dimensional layer of sp(2)-hybridized carbon atoms, can be viewed as a sheet of benzene rings fused together. Three benzene rings can be combined in three different ways, to yield linear anthracene and angular phenanthrene, where the rings share two C-C bonds, and the phenalenyl structure where three C-C bonds are shared between the rings. This third structure contains an uneven number of carbon atoms and, hence, in its neutral state, an uneven number of electrons--that is, it is a radical. All three structures may be viewed as being sections of graphene. Extension of this concept leads to an entire family of phenalenyl derivatives--'open-shell graphene fragments'--that are of substantial interest from the standpoint of fundamental science as well as in view of their potential applications in materials chemistry, in particular quantum electronic devices. Here we discuss current trends and challenges in this field.