  The existence, uniqueness, and asymptotic behavior of steady transonic flows
past a curved wedge, involving transonic shocks, governed by the
two-dimensional full Euler equations are established. The stability of both
weak and strong transonic shocks under the perturbation of both the upstream
supersonic flow and the wedge boundary is proved. The problem is formulated as
a one-phase free boundary problem, in which the transonic shock is treated as a
free boundary. The full Euler equations are decomposed into two algebraic
equations and a first-order elliptic system of two equations in Lagrangian
coordinates. With careful elliptic estimates by using appropriate weighted
H\"older norms, the iteration map is defined and analyzed, and the existence of
its fixed point is established by performing the Schauder fixed point argument.
The careful analysis of the asymptotic behavior of the solutions reveals
particular characters of the full Euler equations.
