Chapter 1 introduces the most important terms and concepts of cardiovascular physiopathology ,while Chapter 2 illustrates the basic mathematical models for blood flow and biochemical transfer. The derivation of the equations that governs blood flow is covered in Chapter 3, while Chapter 4 is devoted to the treatment of medical images to obtain geometries suitable for numerical computations. Chapter 5 illustrates the important relationship between geometry and type of flow, focusing on the main characteristics of the different flow regimes encountered in the cardiovascular system. Mathematical models for blood rheology are discussed in Chapter 6. In Chapter 7 mathematical and numerical models of biochemical transport are explained in detail, with practical examples. The mathematical analysis of coupled models for fluid-structure interaction is addressed in Chapter 8, while Chapter 9 focuses on numerical methods for the mechanical coupling between blood flow and the vessel structure. Reduced models play an important role in cardiovascular modelling to enable the simulating of large parts of (or even th whole) vascular system. Their derivation is presented in Chapter 10. The intertwining of such models with more complex three dimensional ones is the foundation of the so called geometric multiscale approach, illustrated in detail in Chapter 11. Finally, Chapter 12 provides a set of well described and reproducible test cases and applications.