This paper proves several Tauberian theorems for general iterations of operators, and provides two applications to zero-sum stochastic games where the total payoff is a weighted sum of the stage payoffs. The first application is to provide conditions under which the existence of the asymptotic value implies the convergence of the values of the weighted game, as players get more and more patient. The second application concerns stochastic games with finite state space and action sets. This paper builds a simple class of asymptotically optimal strategies in the weighted game, that at each stage play optimally in a discounted game with a discount factor corresponding to the relative weight of the current stage. models (see for instance the recent surveys [5], [2] and [18]). A natural question is to ask whether there is a link between the convergence of (v n ), (v λ ) and (v θ ) * . Ziliotto [22] has proved that in a very general stochastic game model, with possibly infinitely many states and actions, (v n ) converges uniformly if and only if (v λ ) converges uniformly † . This paper aims at generalizing such a result to a more general family of values (v θ ). Note that for recursive games (stochastic games in which the stage payoff is 0 in nonabsorbing states), Li and Venel [7] have proved that if (v n ) or (v λ ) converges uniformly, then (v θ ) converges uniformly when θ is decreasing and θ 1 goes to 0. In a dynamic programming framework (one player), Monderer and Sorin [9] have proved the convergence property, but for a more restricted class of decreasing evaluations. The contribution of this paper is twofold. In the same stochastic game model as in [22], it provides conditions under which the uniform convergence of (v n ) or (v λ ) implies the uniform convergence of (v θ ). Second, in the case where the state space and the action sets are finite, it proves that the following discounted strategy is asymptotically optimal in the θ-weighted game, as θ 1 goes to 0: at each stage m ≥ 1, play optimally in the discounted game with discount factor θ m /( m ′ ≥m θ m ′ ) ‡ . Such a discount factor corresponds to the weight of stage m relative to the weight of future stages. This result is new even when θ is a n-stage evaluation. Finally, this paper provides an example that illustrates the sharpness of the first result.The proof of the results rely on the operator approach, introduced by Rosenberg and Sorin [16]. This approach builds on the fact that the value of the θ-weighted game satisfies a functional equation, called the Shapley equation (see [17]). The properties of the associated operator can be exploited to infer convergence properties of (v θ ) (see [16]). This paper first proves several Tauberian theorems in an operator setting, and apply them to stochastic games to get the first result. Surprisingly, the proof of the second result also follows from a Tauberian theorem for operators. It is due to the fact that the payoff w θ guaranteed by a discounted strategy in the θ-weighted game satisfies a functional equatio...