This dissertation establishes and investigates new phenomena in diverse interacting many-body quantum systems guided by three distinct, but complementary, themes: (i) symmetry and topology, (ii) localization, and (iii) non-Fermi liquids.

The first theme concerns how the interplay of symmetry and topology can offer robust protection for a many-body system. We investigate low-dimensional quantum fermionic models from a general structural perspective. These phases can exhibit fractionalized Majorana zero-energy modes on their boundary. We devise experimentally relevant nonlocal measurements that can be used to detect these topological phases. While our primary focus is on quantum systems, topologically protected behavior can arise in classical mechanical models as well. We extend a recent connection between the topological band theory of electrons and classical physics by proposing a mechanical analogue of a topological nodal semimetal.

The second theme concerns that of many-body localization. We demonstrate that the combination of localization, symmetry, and topology can have radical consequences for quantum systems at high energies, such as the existence of protected gapless boundary modes. We show that, even at these high energies, quantum information can be preserved, and quantum coherence recovered. Quantum coherent dynamics in this regime is unexpected and of great interest for quantum computation.

The third direction in our study of interacting many-body systems concerns non-Fermi liquids. We construct a non-Fermi liquid by bringing together a spin-orbit coupled Fermi surface and fluctuating magnetic order. Using newly developed analytic tools for strongly coupled systems, we demonstrate the stability of the non-Fermi liquid to ordering. This identifies an experimentally accessible candidate for exploring physics that lies beyond Fermi liquid theory.