Vorticity formulations of the incompressible Navier-Stokes equations circumvent the challenges associated with the pressure-velocity numerical coupling. Furthermore, vorticity dynamics in challenging fluid flow problems can be analysed more effectively within a vorticity-based numerical framework. However, deriving suitable vorticity boundary conditions for numerical simulations with high-order accuracy on curved domains remains a significant challenge for these formulations, particularly in three dimensions. Consequently, most existing work considers only rectangular domains with regular grids, limiting its application to geometries of practical interest. This study presents suitable wall vorticity boundary conditions for incompressible vorticity equations on arbitrary curved domains, introducing a simple, effective, and highly accurate numerical treatment based on a finite volume discretisation on unstructured meshes. Moreover, a novel technique is employed to discretise the boundary conditions on a linear piecewise approximation of the curved boundary, circumventing the challenges associated with curved meshes while preserving very high-order accuracy. The two-dimensional streamfunction-vorticity formulation is analysed both theoretically and numerically, but these advancements establish a foundation for other vorticity-based formulations and the three-dimensional case, in which the treatment of vorticity boundary conditions remains fundamentally unchanged. To demonstrate the accuracy and effectiveness of the proposed methodology, several benchmark incompressible fluid flow problems in non-trivial two-dimensional curved domains are presented, including the semi-elliptical lid-driven cavity problem, for which very high orders of convergence are achieved without any regularisation.