Our prolonged interest in the transonic wedge/cone flow problems stems from our earlier pursuit of Oswatitsch's parabolic method [1-3]. The intricate nonlinearity imbedded in the subsequent improved parabolic methods, such as localinearization and nonlinear-correction [4,5], motivates our continuous study of the transonic small disturbance equation (TSDE) through the rather different approach developed in . The first part of this paper thus focuses on this technique, revisiting it in the context of wedges and further extending it to cone flows. The second part of the paper addresses another transonic problem, i.e. wedges supporting attached curved shocks. To this end, a perturbed Euler's equations formulation and its first-order results are presented.