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Our main result is to establish stability of martingale couplings: supposethat $\pi$ is a martingale coupling with marginals $\mu, \nu$. Then, givenapproximating marginal measures $\tilde \mu \approx \mu, \tilde \nu\approx \nu$in convex order, we show that there exists an approximating martingale coupling$\tilde\pi \approx \pi$ with marginals $\tilde \mu, \tilde \nu$.  In mathematical finance, prices of European call / put option yieldinformation on the marginal measures of the arbitrage free pricing measures.The above result asserts that small variations of call / put prices lead onlyto small variations on the level of arbitrage free pricing measures.  While these facts have been anticipated for some time, the actual proofrequires somewhat intricate stability results for the adapted Wassersteindistance. Notably the result has consequences for a several related problems.Specifically, it is relevant for numerical approximations, it leads to a newproof of the monotonicity principle of martingale optimal transport and itimplies stability of weak martingale optimal transport as well as optimalSkorokhod embedding. On the mathematical finance side this yields continuity ofthe robust pricing problem for exotic options and VIX options with respect tomarket data. These applications will be detailed in two companion papers.

Approximation of martingale couplings on the line in the weak adapted topology