Isometric immersion is an important problem in geometry with emerging applications. Its connection with transonic flows in fluid dynamics will be discussed. The Gauss-Codazzi equations for isometric immersions of surfaces will be considered. Various approaches and recent results on global solutions for the hyperbolic problem with negative Gauss curvatures will be presented. In particular, the global smooth isometric immersion of surfaces with slow decay rates for the curvatures will be discussed.