  Understanding the far-from-equilibrium dynamics of dissipative quantum
systems, where dissipation and decoherence coexist with unitary dynamics, is an
enormous challenge with immense rewards. Often, the only realistic approach is
to forgo a detailed microscopic description and search for signatures of
universal behavior shared by collections of many distinct, yet sufficiently
similar, complex systems. Quantum chaos provides a powerful statistical
framework for addressing this question, relying on symmetries to obtain
information not accessible otherwise. This thesis examines how to reconcile
chaos with dissipation, proceeding along two complementary lines. In Part I, we
apply non-Hermitian random matrix theory to open quantum systems with Markovian
dissipation and discuss the relaxation timescales and steady states of three
representative examples of increasing physical relevance: single-particle
Lindbladians and Kraus maps, open free fermions, and dissipative
Sachdev-Ye-Kitaev (SYK) models. In Part II, we investigate the symmetries,
correlations, and universality of many-body open quantum systems, classifying
several models of dissipative quantum matter. From a theoretical viewpoint,
this thesis lays out a generic framework for the study of the universal
properties of realistic, chaotic, and dissipative quantum systems. From a
practical viewpoint, it provides the concrete building blocks of dynamical
dissipative evolution constrained by symmetry, with potential technological
impact on the fabrication of complex quantum structures.
  (Full abstract in the thesis.)
