Quantum affine algebras are quantum enveloping algebras of affine Lie algebras, introduced independently by Drinfeld and Jimbo in their study of the Yang-Baxter equation. Representation theory of quantum affine algebras depends mostly on the Drinfeld realization, which was announced in 1987. Ding and Frenkel proved the presentation using the Faddeev school's R-matrix method in type A in 1993. We will discuss our recent joint work with A. Molev and M. Liu on the Drinfeld realization for other classical types. Our proof extended the R-matrix method with the help of quasideterminants, and we constructed a sequence of larger quantum algebras associated with the spectral parameter dependent R-matrix. In the end, I will also briefly discuss another presentation of quantum extended affine algebras (jointly with F. Chen, F. Kong and S. Tan).