MUCH attention has recently been given to the investigation of transonic problems of gas dynamics. Such flows occur, for instance, in nozzles, in the circulation round bodies with a receding shock wave, in the development of the flow about a break point forming a convex angle, and also on aerofoils with local supersonic regions, etc. Aerodynamic and stability characteristics and questions of control and stability of motion have clearly been insufficiently studied under transonic conditions. Difficulties arise in the theoretical consideration of such problems firstly because in a number of cases there is no complete mathematical formulation of the problem. In regions of mixed gas flows complicated phenomena take place. Behind the main shock wave so-called secondary “suspended” shock waves may occur and in the critical circulation round bodies local supersonic regions are formed which may also be closed by shock waves, etc. The mechanism by which such phenomena occur has not been sufficiently studied and the question of the form of the minimum region of influence and its reconstruction with change of the circulation parameters etc. also remains open. It must also be remarked that for rotational plane and three-dimensional transonic problems no analytical and numerical methods have been developed in practice. Classical analytical methods, developed for plane potential flows and based principally on the use of the hodograph plane and a different simplification and linearization, do not as a rule apply here. Numerical solution of the problem is very difficult and requires the construction of special schemes for the integration of elliptico-hyperbolic equations. In this paper we consider some problems of transonic gas-dynamics related to the theory of plane space rotational flows of an ideal gas, worked out basically by the authors or under their direction. The first part of the paper gives the results of the analytical investigation of a number of exact properties of solutions of proplems of the circulation round bodies with a receding shock wave. The results obtained are given without proof in the forms of theorems (the proofs are in the listed works or will be published in the near future). The second part considers numerical schemes and gives results of circulation for the solution of the complete problem of the supersonic circulation round blunt bodies with a break in the generating contour, and also at low supersonic speeds. We also study here the limits of existence of various types of minimum regions of influence of a mixed flow with blunting. Only numerical schemes using high-speed computers and carefully conducted experiments enable us to obtain quantitative data and a complete picture of the flow in such complicated problems. Here we make an attempt to examine and discuss some properties of transonic flows (solutions in the neighbourhood of the sonic line, the formation of secondary shock waves, the form of the minimum zone of influence etc.) with the positions of the analytical and numerical solutions. It should be pointed out that by this numerical method it is possible to discover and construct secondary shock waves, non-monotonic sonic lines, etc. At the same time the conditions for the collapse of a continuous flow and rules connected with various types of regions of influence were discovered analytically. In the consideration of flows round an angle the effective construction of numerical schemes turns out to be possible only by using asymptotic methods of solving differential equations, etc. Thus the amalgamation of both methods appears to be very fruitful. This work is extended in these directions.