A method is proposed for the adaptive refinement and coarsening of two-dimensional meshes consisting of arbitrarily polygonal cells, which are extensively used in unstructured finite-volume CFD software. It generalizes a previous implementation of the method for 3D meshes [1, 2], motivated by extensive use of two-dimensional meshes for situations that balance computational cost with efficiency. Its implementation in ANSYS Fluent has been shown to suit a large variety of applications, ranging from multiphase flows to shock-capturing external aerodynamics. The scheme is also compatible with prismatic boundary layer refinement, which is often necessary to resolve near-wall turbulent flows. This work will explain the methodology behind the approach along with examples, with emphasis on cell-quality constraints and parallel efficiency.