During the last years adoption of Scientific Computing tools in the investigation of Cardiovascular Diseases has formidably increased. Manipulation of medical images and data through mathematical models allows to understand physiopathological dynamics in the single patient where numerical simulations of blood flow is nowadays possible. Moreover, prediction of the effects of a surgery or a therapy are made possible by the numerical solution of partial differential equations. Complexity of the problems at hand in this field demands however for specific and effective numerical techniques. Blood flow problems are unsteady, heterogeneous (fluid+structure+chemical dynamics+electrodynamics), multi-scale in time and space. These features require often ad hoc mathematically sound techniques in order to have reliable results in reasonable computational times. In this talk, I will consider some examples stemming from different collaborations with medical doctors, covering the most challenging mathematical problems we found in this context.