Keywords: Blood flow, microcirculation, modeling, rheology Abstract: Various rich dynamics of vesicles under linear and nonlinear flows will be discussed. We present analytical and numerical results on tank-treading motion, tumbling and vacillating-breathing (aka swinging, trembling). We then discuss the notion of transverse migration due to a wall and to a nonlinear flow. We show theoretical and exeprimental results on the law of transverse migration in a microfluidic device. Finally, we present very recent results on a longstadning puzzle of the blood microcirculatory research: why do red blood cells adopt a non symmetric shape (called slipper) in small blood vessels? A key result of our study is that the parachute symmetric shape is shown to be unstable, while the slipper shape is stable. That is, small flow disturbances–which are always present in real blood flows–cause RBCs to assume slipper shapes. It is further shown that the slipper shape offers a better transport efficiency to RBCs. In addition the slipper shape favors hemoglobin mixing in the cell, and thus enhances oxygen transport efficiency. Blood flow efficiency together with optimal oxygen supply seem to be determinant for natural selection of slipper shapes.