In this talk, we first consider the derivative nonlinear Schrödinger equation (DNLS) on the torus. By employing the Fourier restriction norm method, the $I$-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao, the normal form method, and a coercivity property in the spirit of Guo and Wu, we prove global well-posedness in $H^{1/2}(\mathbb{T})$, provided initial data with mass less than $4\pi$. 　Then, we consider DNLS on the real-line and study the uniqueness of its solutions. In particular, by implementing an infinite iteration of the normal form reductions based on a scheme introduced by Guo, Kwon, and Oh, we prove unconditional uniqueness for DNLS in $H^s(\mathbb{R})$, $s>1/2$. The second part is joint work with Haewon Yoon (Korea Advanced Institute of Science and Technology).