We present a new R-adaptive Arbitrary Lagrangian Eulerian (ALE) method, based on the reconnection-based ALE - ReALE methodology [5, 41, 42]. The main elements in a standard ReALE method are: an explicit Lagrangian phase on an arbitrary polygonal (in 2D) mesh, followed by a rezoning phase in which a new grid is defined, and a remapping phase in which the Lagrangian solution is transferred onto the new grid. The rezoned mesh is smoothed by using one or several steps toward centroidal Voronoi tessellation, but it is not adapted to the solution in any way. We present a new R-adaptive ReALE method (R-ReALE, where R stands for Relocation). The new method is based on the following design principles. First, a monitor function (or error indicator) based on Hessian of some flow parameter(s), is utilized. Second, the new algorithm uses the equidistribution principle with respect to the monitor function as criterion for defining an adaptive mesh. Third, centroidal Voronoi tessellation is used for the construction of the adaptive mesh. Fourth, we modify the raw monitor function (scale it to avoid extremely small and large cells and smooth it to create a smooth mesh), in order to utilize theoretical results related to centroidal Voronoi tessellation. In the R-ReALE method, the number of mesh cells is chosen at the beginning of the calculation and does not change with time, but the mesh is adapted according to the modified monitor function during the rezone stage at each time step. We present all details required for implementation of the new adaptive R-ReALE method and demonstrate its performance relative to standard ReALE method on a series of numerical examples.