The talk will summarize some results relating to clarifying the physical significance of the characteristic structure of the Weyl curvature tensor, and proposals for its utilization. I will begin by showing how null force-free or vacuum electrodynamic solutions experience reduced scattering by propagating along the principal null directions (GPNDs) of the spacetime, as if they were the flatter directions of the curvature tensor. Building on this observation, I will use complexified sectional curvatures to develop an analogy between the Weyl tensor and the second fundamental form of an immersed surface, and argue that, despite their name, the GPNDs are more akin to the asymptotic directions rather than the principal directions. I will also identify the ``true'' principal directions and curvatures, which has so far been under-utilized. I will then move on to propose usage for these quantities in topics related to the peeling theorem, the large scale geometry of the spacetime, the transition from the merger phase of binary black hole merger into the quasinormal mode ringdown phase, and perturbation of generic background spacetimes.