Considerable confusion surrounds the longstanding question of whatconstitutes a vortex, especially in a turbulent flow. This question,frequently misunderstood as academic, has recently acquiredparticular significance since coherent structures (CS) in turbulentflows are now commonly regarded as vortices. An objective definitionof a vortex should permit the use of vortex dynamics concepts toeduce CS, to explain formation and evolutionary dynamics of CS, toexplore the role of CS in turbulence phenomena, and to developviable turbulence models and control strategies for turbulencephenomena. We propose a definition of a vortex in an incompressibleflow in terms of the eigenvalues of the symmetric tensor${\bm {\cal S}}^2 + {\bm\Omega}^2$; here ${\bm {\cal S}}$ and ${\bm \Omega}$ are respectively the symmetricand antisymmetric parts of the velocity gradient tensor${\bm \Delta}{\bmu}$. This definition captures the pressure minimumin a plane perpendicular to the vortex axis at high Reynoldsnumbers, and also accurately defines vortex cores at low Reynoldsnumbers, unlike a pressure-minimum criterion. We compare ourdefinition with prior schemes/definitions using exact and numericalsolutions of the Euler and Navier–Stokes equations for a variety oflaminar and turbulent flows. In contrast to definitions based on thepositive second invariant of ${\bm\Delta}{\bm u}$ or the complex eigenvalues of${\bm \Delta}{\bmu}$, our definition accurately identifies thevortex core in flows where the vortex geometry is intuitivelyclear.