Solitons and breathers are localized solutions of integrable systems that can be viewed as "particles'' of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media these ``integrable'' gases present fundamental interest for nonlinear physics. We develop nonlinear spectral theory of breather and soliton gases by considering a special, thermodynamic type limit of the nonlinear dispersion relations for multi-phase (finite-gap) solutions of the focusing nonlinear Schrödinger (fNLS) equation. A number of concrete examples of breather and soliton gases are considered, demonstrating efficacy of the developed general theory and also having some interesting implications. In particular, the statistical properties of a special kind of soliton gas, that we term the bound state soliton condensate, reveal a remarkable connection with the nonlinear stage of modulational instability. This is joint work with Alex Tovbis (Central Florida).