The work addresses an important question of whether a discontinuity in wall curvature can cause boundary layer separation at transonic speeds. Firstly an inviscid transonic flow in the vicinity of a curvature break is analysed. Depending on the ratio of the curvatures, several physically different regimes can exist, including a special type of supersonic flows which decelerate to subsonic speeds without a shock wave, transonic Prandtl-Meyer flow and supersonic flows with a weak shock. It is shown that if the flow can be extended beyond the limiting characteristic, it subsequently develops a shock wave. As a consequence, a fundamental link between the local and the global flow patterns is observed in our problem. From an asymptotic analysis of the Karman-Guderley equation it follows that the curvature discontinuity leads to singular pressure gradients upstream and downstream of the break point. To find these gradients, we perform computations and employ both the hodograph method and the phase portrait technique. The focus is then turned to analysing how the given pressure distribution affects the boundary layer. It is demonstrated that the singular pressure gradient, which appears to be proportional to the inverse cubic root of the distance form the curvature break, corresponds to a special resonant case for the boundary layer upstream of the singularity. Consequently, the boundary layer approaches the interaction region in a pre-separated form. This changes the background on which the viscous-inviscid interaction develops, allowing to construct an asymptotic theory of the incipient viscous-inviscid interaction for our particular problem. The analysis of the interaction which takes place near a weak curvature discontinuity leads to a typical three-tier structure. It appears to be possible to obtain analytical solutions in all the tiers of the triple deck when the curvature break is small. As a result, the interaction equation may be derived in a closed form. The analytical solution of the interaction equation reveals a local minimum in the skin friction distribution, suggesting that a local recirculation zone can develop near the curvature break. In fact, the recirculation zone is formed when the ratio of the curvatures is represented as a series based on negative powers of the logarithm of the Reynolds number. This proves that a discontinuity in wall curvature does evoke boundary layer separation at transonic speeds. The result is fundamentally different from the effect of a curvature break at subsonic and supersonic speeds, as no separation takes place in these two regimes (Messiter & Hu 1975).