Swarm Intelligence (SI) Algorithms imitate the collective behavior of various swarms or groups in nature. In this work, three representative examples of SI algorithms have been selected and thoroughly described, namely the Grey Wolf Optimizer (GWO), the Whale Optimization Algorithm (WOA), and the Salp Swarm Algorithm (SSA). Firstly, the selected SI algorithms are reviewed in the literature, specifically for optimization problems in antenna design. Secondly, a comparative study is performed against widely known test functions. Thirdly, such SI algorithms are applied to the synthesis of linear antenna arrays for optimizing the peak sidelobe level (pSLL). Numerical tests show that the WOA outperforms the GWO and the SSA algorithms, as well as the well-known Particle Swarm Optimizer (PSO), in terms of average ranking. Finally, the WOA is exploited for solving a more computational complex problem concerned with the synthesis of an dual-band aperture-coupled E-shaped antenna operating in the 5G frequency bands.