Transonic flows are important mixed-type problems with a broad range of applications and present challenges in analysis. In this talk, I will first briefly review the basic theory of the Euler equations in gas dynamics. Then I will present results on transonic flows past obstacles, transonic flows within the fluid dynamic formulation of isometric embeddings in geometry, and transonic flows in nozzles. I will discuss global solutions and stability achieved through various techniques and approaches. Outstanding open problems in this area will also be discussed.