The goal of this dissertation is to advance the theoretical understanding of complex electromagnetic systems by establishing newly discovered performance bounds for several critical problems that are active areas of research. These bounds are derived from basic physics constraints such as causality, energy conservation, and passivity. In particular, the analysis focuses on planar architectures, such as metasurfaces and thin-film platforms, which are compatible with standard semiconductor processes and thus enable scalable production and device miniaturization. These characteristics are crucial for future applications that increasingly demand smaller footprints and enhanced performance. The problems investigated include: (i) the maximum achievable reflected power from a thin planar structure across all reflection angles and polarizations; (ii) the maximal Q-factor of planar inductors within a prescribed footprint; (iii) the broadband limits of giant nonreciprocal Faraday rotation in magnetized plasma materials; and (iv) fundamental constraints on invisibility cloaks in light of a recent proposal involving fast-light media. The derived bounds serve as both theoretical limits and practical benchmarks for evaluating and guiding design efforts. Comparisons with state-of-the-art designs from the literature highlight opportunities for further performance improvements. It is hoped that this dissertation will encourage researchers and engineers to place greater emphasis on understanding the fundamental limitations inherent in their design problems, thereby complementing inverse design algorithms to realize electromagnetic devices with maximal achievable performance at reduced computational effort.