We reveal that under moderate shear stress (ηγ ≈ 0.1 Pa) red blood cells present an oscillation of their inclination (swinging) superimposed to the long-observed steady tanktreading (TT) motion. A model based on a fluid ellipsoid surrounded by a visco-elastic membrane initially unstrained (shape memory) predicts all observed features of the motion: an increase of both swinging amplitude and period (1/2 the TT period) upon decreasing ηγ, a ηγ-triggered transition towards a narrow ηγ-range intermittent regime of successive swinging and tumbling, and a pure tumbling motion at lower ηγ-values. A human red blood cell (RBC) is a biconcave flattened disk, essentially made of a Newtonian hemoglobin solution encapsulated by a fluid and incompressible lipid bilayer, underlined by a thin elastic cytoskeleton (spectrin network) [1]. The complex structure of RBCs and their response to a viscous shear flow have a great influence on flow and mass transport in the microcirculation in both health and disease [2]. The full understanding of this response requires a direct comprehensive observation of cell motion and deformation, and a model for deducing the cell intrinsic properties from its behavior in shear flow. It is generally admitted that the two possible RBC movements are the unsteady tumbling solid-like motion [3], and the drop-like 'tanktreading' motion for higher shear stresses, where the cell maintains a steady orientation, while the membrane rotates about the internal fluid, as reported respectively for RBCs suspended in plasma or in high-viscosity media and submitted to high shear stresses [3,4,5,6]. However, the RBC movement at smaller shear rate and close to the tumbling-tanktreading transition, has not been fully explored. Moreover, the actual state of deformation of the elastic skeleton either in the flowing or in the resting RBC is still conjectural ("shape memory" problem) [7]. Most models [6,8] derive from the analytical framework of Keller and Skalak (KS) [9], which treats the RBC as a fluid ellipsoidal membrane enclosing a viscous liquid. Although this model qualitatively retrieves the two modes of motion, it does not capture the observed shear-rate dependency of the tumblingtanktreading transition. In particular, the model does not account for the possible elastic energy storage induced by the local deformations of the cytoskeleton during tanktreading. Approaches including membrane elasticity are either restricted to spherical resting shapes because of analytical complexities [10], or propose encouraging but still limited numerical analysis on tanktreading elastic biconcave capsules [11].Here, we reveal a new regime of motion for RBCs under small shear flow, characterized by an elastic capsule-like oscillation of the cell inclination superimposed to tanktreading that we name swinging. We develop a model, which predicts both swinging and the shear-stress dependency of the tumbling-tanktreading transition. It demonstrates the existence of the elastic shape memory in the membrane.Direct measurements of cell ori...