  Let $\textbf{U}^+$ be the positive part of the quantum group $\textbf{U}$
associated with a generalized Cartan matrix. In the case of finite type,
Lusztig constructed the canonical basis $\textbf{B}$ of $\textbf{U}^+$ via two
approaches. The first one is an elementary algebraic construction via
Ringel-Hall algebra realization of $\textbf{U}^+$ and the second one is a
geometric construction. The geometric construction of canonical basis can be
generalized to the cases of all types. The generalization of the elementary
algebraic construction to affine type is an important problem. We give several
main results of algebraic constructions to the affine canonical basis in this
ariticle. These results are given by Beck-Nakajima, Lin-Xiao-Zhang,
Xiao-Xu-Zhao, respectively.
