In this paper, we study a transonic shock problem for the Euler flows through a class of 2-D or 3-D nozzles. The nozzle is assumed to be symmetric in the diverging (or converging) part. If the supersonic incoming flow is symmetric near the divergent (or convergent) part of the nozzle, then, as indicated in Section 147 of [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publ., New York, 1948], there exist two constant pressures P1 and P2 with P1<P2 such that for given constant exit pressure Pe ∈ (P1, P2), a symmetric transonic shock exists uniquely in the nozzle, and the position and the strength of the shock are completely determined by Pe. Moreover, it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting. Furthermore, we establish the global existence, stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure Pe when the initial unsteady shock lies in the symmetric diverging part of the 2-D or 3-D nozzle. On the other hand, it is shown that an unsteady symmetric transonic shock is structurally unstable in a global-in-time sense if it lies in the symmetric converging part of the nozzle.