  Quantum universal enveloping algebras, quantum elliptic algebras and double
(deformed) Yangians provide fundamental algebraic structures relevant for many
integrable systems. They are described in the FRT formalism by R-matrices which
are solutions of elliptic, trigonometric or rational type of the Yang--Baxter
equation with spectral parameter or its generalization known as the
Gervais--Neveu--Felder equation. While quantum groups and double Yangians
appear as quasi-triangular Hopf algebras, this is no longer the case for
elliptic algebras and the various deformations of Yangian type algebras. These
structures are dealt with the framework of quasi-Hopf algebras. These algebras
can be obtained from Hopf algebras through particular Drinfel'd twists
satisfying the so-called shifted cocycle condition. We review these different
structures and the pattern of connections between them.
