The self-dual Chern-Simons-Schrödinger model is a gauged cubic NLS on the plane with self-duality, i.e., energy minimizers are given by a first-order Cauchy-Riemann-type equation, rather than a second-order elliptic equation. While this equation shares all formal symmetries with the usual cubic NLS on the plane, the structure of solitary waves is quite different due to self-duality and nonlocality (which stems from the gauge structure). In accordance, this model possesses blow-up and global dynamics that are quite different from that of the usual cubic NLS. The goal of this talk is to present some recent results concerning the blow-up and global dynamics of this model, with emphasis on a few surprising features of this model such as the impossibility of a "bubble-tree'' blow-up and a nonlinear rotational instability of pseudoconformal blow-ups. This talk is based on joint work with Kihyun Kim (IHES) and Soonsik Kwon (KAIST).