  Various aspects of the theory of quantum integrable systems are reviewed.
Basic ideas behind the construction of integrable ultralocal and nonultralocal
quantum models are explored by exploiting the underlying algebraic structures
related to the Yang-Baxter equations. Physical meaning of abstract mathematical
notions like universal R-matrix, quantized algebras, Sklyanin algebra, braided
algebra, Hopf algebra etc. and the role played by them in integrable systems
are highlighted. Systematic construction of quantum integrable lattice as well
as field models and their exact excitation spectra are presented through
examples. The coordinate and algebraic formulations of the Bethe ansatz are
illustrated with comparison, along with the description of nested and
functional Bethe ansatzes. The techniques for deriving quantum Hamiltonians
from the Lax operators are demonstrated on concrete models. The exposition of
this review is kept in a fairly elementary level with emphasis on the physical
contents.
