Integrable systems and 1+1 field theories on elliptic curves and its degenerations

We review the classification of elliptic integrable systems described by means of Lax equations and/or Zakharov-Shabat equations. In the finite-dimensional case the classification includes the Calogero-Ruijsenaars family of many-body systems, their spin generalizations, integrable tops, the Gaudin model/spin chain type systems and interacting integrable tops. These models can be extended to 1+1 field theories. Trigonometric and rational degenerations are included into consideration through the usage of R-matrix satisfying the associative Yang-Baxter equation. Interrelations between different type models are discussed as well.