In this paper we find and study the class of symmetric methods among the Runge-Kutta formulae. It is shown that the explicit Runge-Kutta methods cannot be symmetric. We also define the conditions which coefficients in the implicit Runge-Kutta method should satisfy for it to be symmetric. Particular attention has been given to the study of stability properties in the symmetric Runge-Kutta formulae. It is proved that in some cases the notions of absolute and algebraic stability for the given class of numerical methods coincide. Besides, we find a restriction to the order of stable symmetric methods among the diagonally implicit Runge-Kutta formulae. Finally, we give full characteristics of all algebraically stable symmetric Runge-Kutta methods in terms of a transformed matrix of coefficients.