The monography offers a detailed analysis of the methods for constructing mathematical models of transient non-isothermal flows of gas mixtures, multicomponent fluids, and gas - liquid fluids through systems of long branched pipelines including annular sections. To enhance the presentation clarity, the proposed methods are extended to the modeling of flows of heat-conducting multicomponent liquids through long branched open channels. Together with the model construction methods, the strategies of their numerical analysis are discussed in detail, with the use of various classes of finite-difference schemes, including completely conservative spline schemes of high approximation order. Special emphasis is placed on expounding the method of Lagrangian particles as applied to the analysis of heat-conducting multicomponent liquids transmitted through open channels and to the study of complex flows in linear and annular pipeline networks. The methods for constructing...