Complex systems are often described by simple equations that nevertheless can lead to a rich variety of disparate solutions. Discovering and understanding the full spectrum of solutions that correspond to stable states is at the forefront of current research on the network modeling of complex systems. Recent work by our group and others has revealed a myriad of stable and metastable states with remarkable properties, which were once thought to be impossible. In this presentation, I will discuss symmetric states requiring system asymmetry, coherent domains requiring co-existence with incoherent ones, and states remotely synchronized by uncorrelated media. I will motivate the presentation with examples from diverse domains---including metamaterials, microfluidics, physical cryptography, biophysics, and control---and substantiate the main results with both theory and experiments. The presentation will include a discussion of applications and implications for fundamental physics as well as biological and technological systems.