The advantages of turbocharging I.C. Engines have been widely shown in the other chapters, and give clear reasons for the widespread application of this technique in all major applications (from automotive and propulsion to stationary generation, both diesel and SI). However, it is a matter of fact that turbocharged engines are nowadays very complex systems with an overall behavior that is the combined result of system layout, components sizes, control devices and management strategies. All these aspects have to be thoroughly optimized in the design and testing of a turbocharged engine to fulfil the actual tight goals in term of fuel consumption, pollutant emissions and performance. To cope with the intricate task of matching the engine with the turbocharger(s), mathematical models can be very helpful, and for this reason in the last decades they have been playing an increasing role (as in many other applications). ’Virtual’ tools have been assuming more and more significant functions since the first steps of powertrain design leading the way to better overall performance with lower development time and costs. Mathematical models proposed in the scientific literature and used by engineers in the design of turbocharged engines range from very complex 3D models (usually limited to the simulation of single components) to simpler 1D and to even simpler 0D tools for the simulation in Real-Time of the behavior of the whole system. In this chapter, approaches used to build up mathematical models of turbochargers and turbocharged engines are shortly recalled to highlight their scopes within the design process. Particular emphasis will be given to zero-dimensional (0D) models for the Real-Time simulation of turbocharged engines both in steady and in transient operating conditions. Methods and criteria to build them up assembling components sub-models will be described taking account of their applications in the development of the engine and of related management and control system.