In this talk, I will discuss some progress on multi-dimensional steady compressible flows which are some of major
challenges in the theory of multi-dimensional conservations laws in nozzles with variable cross sections.

We will focus on flows with shocks in general variable nozzles with physical boundary conditions.  In particular, I will present some results on the Courant-Friedrich's transonic shock problem in some classes of  general 2-dimensional  nozzles. This will be a nonlinear free boundary value problem with nonlinear boundary conditions for mixed type equations. Existence of single or multiple transonic shocks will be discussed in terms of the geometry of the nozzle and the given exit pressure. Some key ideas of analysis will be presented.  Some open questions will be discussed.