High-resolution and high-cadence solar observations from the ground and from space have revealed the existence of abundant vortex (swirling) motions observed both in quiet and active regions and at all layers of the solar atmosphere. Such motions are present across a broad range of temporal and spatial scales, result mainly from the interaction between turbulent convection and magnetic fields and play a crucial role in the plasma dynamics at the different solar atmospheric layers, hence providing a mechanism for channelling energy from the photosphere up to the low corona. This ISSI international team project aims to bring together world-leading experts in plasma physics and in solar physics (theory, observations and advanced 3D Magneto-HydroDynamic (MHD) simulations). The team will investigate the detailed physics and dynamics of vortex flows as seen in high-resolution, multi-wavelength observations and predicted from state-of-the-art simulations in an attempt to understand the nature and physics of vorticity in the solar atmosphere, its association to atmospheric structuring, magnetic field and generation/propagation of waves and its implications for chromospheric and coronal heating. This ISSI international team is timely as it will coincide with first-light observations (expected in 2019) from the 4-m Daniel K. Inouye Solar Telescope (DKIST) that is going to be the most powerful solar telescope ever built, opening a new window for the observation of small-scale structures, such as swirls, down to a 20-km spatial resolution and where team members participate in its Critical Science Plan with a Science Use Case concerning the multi-scale nature of vorticity in the solar atmosphere. Advancements in observations, theory and simulations to be discussed during the ISSI team meetings will thus generate exciting new results and new international collaborations in this field.