This dissertation explores the general methodology for designing electromagnetic (EM) systems by combining genetic algorithms (GA) with computational electromagnetic (EM) simulations. The EM problems investigated are broadband and multi-band microstrip antennas, low-profile microwave absorbers and electrically small wire antennas. It is shown that optimized performance can be designed and realized using simple shape control. In addition, novel GA approaches are investigated for more challenging multi-objective problems. The developed methodology is also used to explore performance bounds in complex EM systems. First, the use of GA to design microstrip antenna shapes for broadband and multi-band applications is investigated. A full-wave electromagnetic solver is employed to predict the performance of microstrip antennas with arbitrary patch shapes. A GA with two-point crossover and geometrical filtering is implemented to optimize the patch shape. For broadband application, the optimized patch antenna achieves a four-fold improvement in bandwidth when compared to a standard square microstrip. For multi-band application, the optimized patches show that arbitrary frequency spacing ranging from 1:1.1 to 1:2 can be achieved. Tri-band and quad-band microstrip shapes are also generated and the resulting designs show good operations at the designated frequencies. Second, the use of GA for designing optimal shapes for corrugated coatings under near-grazing incidence is examined. Optimized coating shapes depending on different polarizations are generated. Physical interpretations for the optimized structure are discussed, and the resulting shape is compared to conventional planar and triangular shaped designs. This problem is also extended from the single to multi-objective optimization using the Pareto GA. Optimization results using two different objectives, the height (or weight) of the coating versus absorbing performance, are presented. Finally, this dissertation reports on the use of GA in the design optimization of electrically small wire antennas, taking into account of bandwidth, efficiency and antenna size. To efficiently map out this multi-objective problem, the Pareto GA is implemented with the concept of divided range optimization. An optimal set of designs, trading off bandwidth, efficiency and antenna size, are generated. Several GA designs are built, measured and compared to the simulation. Physical interpretations of the GA-optimized structures are provided, and the results are compared against the well-known fundamental limit for small antennas. Further improvements using other geometrical design freedoms are also discussed.