Natural organisms such as photosynthetic bacteria, algae, and plants employ complex molecular machinery to convert solar energy into biochemical fuel. An important common feature shared by most of these photosynthetic organisms is that they capture photons in the form of excitons typically delocalized over a few to tens of pigment molecules embedded in protein environments of light harvesting complexes (LHCs). Delocalized excitons created in such LHCs remain well protected despite being swayed by environmental fluctuations, and are delivered successfully to their destinations over hundred nanometer length scale distances in about hundred picosecond time scales. Decades of experimental and theoretical investigation have produced a large body of information offering insights into major structural, energetic, and dynamical features contributing to LHCs' extraordinary capability to harness photons using delocalized excitons. The objective of this review is (i) to provide a comprehensive account of major theoretical, computational, and spectroscopic advances that have contributed to this body of knowledge, and (ii) to clarify the issues concerning the role of delocalized excitons in achieving efficient energy transport mechanisms. The focus of this review is on three representative systems, Fenna-Matthews-Olson complex of green sulfur bacteria, light harvesting 2 complex of purple bacteria, and phycobiliproteins of cryptophyte algae. Although we offer more in-depth and detailed description of theoretical and computational aspects, major experimental results and their implications are also assessed in the context of achieving excellent light harvesting functionality. Future theoretical and experimental challenges to be addressed in gaining better understanding and utilization of delocalized excitons are also discussed.