The turbulence statistics and structures of a self-similar adverse pressure gradient turbulent boundary layer (APGTBL) are investigated using direct numerical simulation (DNS) of the flow at the verge of separation. The desired self-similar APG-TBL is achieved by a modification of the far-field velocity boundary condition. The required wallnormal velocity in the far-field to produce the necessary adverse pressure gradient was estimated based on the analytical free-stream streamwise velocity distribution for a flow at the point of separation, and the assumption that the streamlines of the outer flow follow the growth of the boundary layer thickness. The APG-TBL develops over a momentum thickness based Reynolds number upto 12000, and achieves a self-similar region of constant friction coefficient, pressure velocity and shape factor. Turbulence statistics in this region show self-similar collapse by using the scaling of the external velocity and the displacement thickness. In this study, the structure of the APG-TBL is investigated using topological methodology and visualisation techniques for a zero pressure gradient turbulent boundary layer (ZPG-TBL) and for the self-similar APG-TBL. The second invariants of the velocity gradient tensor (VGT), which are representative of coherent structures dominated by vortical motions, show a stark difference in the structure and location of coherent vortical structures that exists between the self-similar APGTBL and a ZPG-TBL. Further details based on the structure and distributions of the invariants of VGT and intense Reynolds stress structures of the self-similar APG-TBL are presented.