Be it to aerate a glass of wine before tasting, to accelerate a chemical reaction or to cultivate cells in suspension, the “swirling” (or orbital shaking) of a container ensures good mixing and gas exchange in a simple and intuitive way. Despite being used in such a large variety of applications, the flow generated in a container subject to orbital shaking is far from be- ing understood, and presents a richness of patterns and behaviours which has not yet been reported. While orbital shaken cell cultures are very efficient and productive at small scale, their increase in scale is hampered by several issues, some of which are thought to have their origin in the motion of the liquid medium. The present research remedies to this situation, charting the evolution of the wave behaviour with the operating parameters, highlighting the importance of the wave regimes and assessing their mixing efficiency. We present here a mathematical solution, based on the potential hypothesis and on techniques used in sloshing dynamics, predicting the shape of the free surface and the liquid motion. The validity and the limits of this model were assessed by comparison with a very large number of free surface measurement, obtained using a specifically developed automated acquisition system, and with non intrusive velocity measurements of several shaking configurations. A large variety of wave patterns (i.e. free surface shapes) were identified, ranging from single and multiple crested waves to breaking waves and waves having a shape constantly changing as they rotate. Our research revealed the importance of free surface natural modes and their sub-harmonics in the behaviour of the waves. From the results of the potential model and the measurements, we identified four dimensionless groups ensuring hydrodynamic similarity of the flow be- tween different scales. Moreover, we were able to identify the most efficient waves in terms of mixing, and to suggest optimal ranges of the operating parameters to enhance the mixing and oxygenation of the cell cultures.