Introduction: The Role and the Development of Computational MethodsThe introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new -clearly noninvasiveapproach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations. In this respect, the terrific development of new devices and algorithms for image and data retrieval and processing allowed the migration from (over)simplified idealized descriptions to high-fidelity patient-specific models, in a critical -still ongoingprocess of merging measures and conceptualizations. Meanwhile, the progressive improvement of computational resources provides the infrastructures to perform numerical simulations of complex dynamics in reasonable timelines. All such processes required, in background, the crucial development of novel specific modeling techniques and solvers in the field of computational mechanics, in an exciting process involving mathematics, engineering, computer science, and biomedical knowledge that is now having an impact not only on research but also on clinical practice. As a matter of fact, mathematical and computational modeling has been recognized more than a research tool, a service to be integrated in products for the clinical market, as the example of the company HeartFlow [Taylor et al., 2013] demonstrates.