Mathematical sciences are contributing more and more to advances in life science research, a trend that will grow in the future. Realizing that the mathematical sciences can be critical to many areas of biomedical imaging, we organized a three-day minicourse on mathema- cal modelling in biomedical imaging at the Institute Henri Poincar´einParis in March 2007. Prominent mathematicians and biomedical researchers were paired to review the state-of-the-art in the subject area and to share mat- maticalinsightsregardingfutureresearchdirectionsinthisgrowingdiscipline. The speakers gave presentations on hot topics including electromagnetic brain activity, time-reversal techniques, elasticity imaging, infrared thermal tomography, acoustic radiationforce imaging, electrical impedance and m- netic resonance electrical impedance tomographies. Indeed, they contributed to this volume with original chapters to give a wider audience the bene't of their talks and their thoughts on the ?eld. This volume is devoted to providing an exposition of the promising - alytical and numerical techniques for solving important biomedical imaging problems and to piquing interest in some of the most challenging issues. We hope that it will stimulate much needed progress in the directions that were described during the course. The biomedical imaging problems addressed in this volume trigger the investigation of interesting and di?cult problems in various branches of mathematics including partialdi?erential equations, h- monic analysis, complex analysis, numerical analysis, optimization, image analysis, and signal theory. The partial support o?ered by the ANR project EchoScan (AN-06- Blan-0089) is acknowledged. We also thank the sta? at the Institute Henri Poincar´ e.