  The aim of this review is to present the list of by now a significant
collection of quantum integrable models, ultralocal as well as nonultralocal,
in a systematic way stressing on their underlying unifying algebraic
structures. We restrict to quantum and statistical models belonging to
trigonometric and rational classes with (2 x 2)- Lax operators. The ultralocal
models can be classified successfully through their associated quantum algebra
and are governed by the Yang-Baxter equation, while the nonultralocal models,
the theory of which is still in the developmental stage, allow systematization
through the braided Yang-Baxter equation. Along with the known integrable
models some possible directions for investigation in this field and generation
of such new models are suggested.
