This work is the result of 10 years of collaboration with clinicians. It is an attempt to both 'bring about [...] an adequate understanding of the other side's language' and to bring new bricks into the 'intricate body of knowledge and skills necessary for effective research progress' [lighthill1975]. Medicine itself is a science that is increasingly becoming quantitative. This new direction calls for sophisticated mathematical models and engineering approaches to unearth deep (patho-)physiological understanding and propose targeted interventions. Such development is supported by recent progress in data acquisition in medicine and biology. The contribution of my work is to bridge frontiers between applied mathematics, bioengineering and biology or medicine. The interactions between these different components are both interesting and challenging. They are challenging, as the fluid mechanics of the application is sometimes so complex that existing numerical methods to solve e.g. the Navier-Stokes equations may be insufficient and necessitate targeted numerical developments (chap. 3). Also the hemo/respiratory in vivo data may not have been acquired with a precision that is high enough to impose coherent boundary conditions in fluid simulations (chap. 4). They are interesting in the sense that applications can drive the development of numerical methods. E.g. blood flow simulations in patient specific geometries and under physiological conditions often led to numerical divergence ten years ago. Ad-hoc strategies first palliated the problem, such as adding more or longer vessels. However this was not always desirable (increase of computational time) or possible (image data resolution), and thus this led the numerical community to revisit stability analysis and stabilization strategies in this context (chap. 3). The physiologically realistic respiratory flow simulations that appeared a few years later than in blood flow, further demonstrated this numerical need. The numerical handling of such complexity in turn made possible for me to simulate pathophysiological conditions that would not have been possible otherwise (chap. 4). In other words, a characteristic of this work is that it is a dynamic loop. The starting point is the goal of answering a biomedical question. This drives the development of mathematical models or numerical tools (applied mathematics/computational mechanics). They are then transferred to the specific application (selection of relevant parameters, inputs from real data, generation of first answer to the biomedical question - bioengineering aspect) to a point where the biomedical question can be addressed (robustness of results assessed based on multiple cases - medicine/biology), thus closing the loop. In this context, my contributions involve adapting or developing models of blood and airflow, at different scales or degrees of precision (chap 2): 3D Navier-Stokes for flow in large conduits, 3D poroelastic formulation compatible with large strain for heart perfusion, 1D equations of blood flow, 0D electric analog for macro or micro-circulation of blood and for respiratory mechanics. Moreover, I have worked on developing numerical methods that are necessary to couple these different models (multidomain or multiscale coupling, with monolithic or robust iterative strategies) and to handle instabilities (contributions in numerical instability analysis, treatment - with or without stabilization - and comparison of different methods, chap. 3). I have also devised strategies to parameterize models from real experimental (animal) or clinical data (depending on the type and amount of data available, based on variational or Kalman filter approaches - chap. 4). Each time, these models and methods are illustrated by a specific biomedical application (contributions in applying these numerical methods to circulation understanding of systemic, in particular coronary, and pulmonary blood circulations, to surgical planning and device design for several congenital heart diseases, to better understanding of emphysema airflow and particle transport in the lung). At the microscale level, my contribution is in the coupling of multiphysics/multiscale systems for modeling the dynamic interplay between tumor cells and their environment, which includes blood perfusion. The manuscript ends with an outlook on topics that necessitate further research (chap 5).