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This work is mainly concerned with the so-called limit theory for mean-fieldgames. Adopting the weak formulation paradigm put forward by Carmona andLacker, we consider a fully non-Markovian setting allowing for drift controland interactions through the joint distribution of players' states andcontrols. We provide first a characterisation of mean-field equilibria asarising from solutions to a novel kind of McKean-Vlasov backward stochasticdifferential equations, for which we provide a well-posedness theory. Weincidentally obtain there unusual existence and uniqueness results formean-field equilibria, which do not require short time horizon, separabilityassumptions on the coefficients, nor Lasry and Lions's monotonicity conditions,but rather smallness, or alternatively regularity, conditions on the terminalreward and a dissipativity condition on the drift. We then take advantage ofthis characterisation to provide non-asymptotic rates of convergence for thevalue functions and the Nash-equilibria of the N-player version to theirmean-field counterparts, for general open-loop equilibria. An appropriatereformulation of our approach also allows us to treat closed-loop equilibria,and to obtain convergence results for the master equation associated to theproblem.

Non-asymptotic convergence rates for mean-field games: weak formulation and McKean-Vlasov BSDEs