Numerical and analytical studies of final states of two-dimensional (2D) decaying turbulence are carried out. The first part of this work is trying to give a definition for final states of 2D decaying turbulence. The functional relation of ¿-¿, which is frequently adopted as the characterization of those final states, is merely a sufficient but not necessary condition; moreover, it is not proper to use it as the definition. It is found that the method through the value of the effective area S covered by the scatter ¿-¿ plot, initially suggested by Read, Rhines, and White ["Geostrophic scatter diagrams and potential vorticity dynamics," J. Atmos. Sci. 43, 3226 (1986)] is more general and suitable for the definition. Based on this concept, a definition is presented, which covers all existing results in late states of decaying 2D flows (including some previous unexplainable weird double-valued ¿-¿ scatter plots). The remaining part of the paper is trying to further study 2D decaying turbulence with the assistance of this definition. Some numerical results, leading to "bar" final states and further verifying the predictive ability of statistical mechanics [Yin, Montgomery, and Clercx, "Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of patches and points," Phys. Fluids 15, 1937 (2003)], are reported. It is realized that some simulations with narrow-band energy spectral initial conditions result in some final states that cannot be very well interpreted by the statistical theory (meanwhile, those final states are still in the scope of the definition).