Leveraging wave matter interactions is central to a myriad of electromagnetic wave-based applications. During the past decades, research on extreme wave manipulation has been revolutionized by artificially engineered materials (metamaterials) and by adding new aspects to the wave-matter interactions that showed intriguing results inaccessible in conventional linear, time invariant (LTI), passive and isotropic media. In this work, I will explore, numerically and experimentally, the possibility of realizing devices that perform beyond or close to their fundamental LTI limitations by adding periodic modulation, nonlinearity, and gain. I will demonstrate these concepts at radio frequencies (RF) and at optical frequencies. Specifically, at RF I ...