  The integrable close and open chain models can be formulated in terms of
generators of the Hecke algebras. In this review paper, we describe in detail
the Bethe ansatz for the XXX and the XXZ integrable close chain models. We find
the Bethe vectors for two--component and inhomogeneous models. We also find the
Bethe vectors for the fermionic realization of the integrable XXX and XXZ close
chain models by means of the algebraic and coordinate Bethe ansatz. Special
modification of the XXZ closed spin chain model ("small polaron model") is
consedered. Finally, we discuss some questions relating to the general open
Hecke chain models.
