Metamaterials are man-made structures developed to engineer electromagnetic waves, and they have been studied intensely in recent decades. Although there are still debates about the performance of these types of artificial materials, many practical applications have been demonstrated using metamaterials from the RF and microwave to IR and optical frequencies. Moreover, due to their unique advantages, they might soon find their path in products and technologies to replace traditional optical and electromagnetic devices. In this dissertation, we primarily study a structure composed of S-shape resonators which show a negative refractive index in a range of frequencies, between 5GHz to 9GHz. We investigate the methods to calculate the constitutive electromagnetic parameters of this metamaterial and offer paths to enhance its performance regarding incident wave polarization. We also propose a meta-substrate to engineer bandwidth of the S-resonator metamaterial. Moreover, we explore properties the proposed metamaterial design for various applications including super-resolution imaging, sensing, and filtering. Planar metamaterials are a 2D version of metamaterials which offer novel applications due to their compact size and low loss. Here, we explore properties of a double ring metasurface for making filters and modulators. Finally, nonlinear effects in metamaterial structures are studied, and a new numerical tool is developed to design metamaterials with natural Two-Photon Absorption nonlinear effect.
