Abstract: We study the problem of designing mechanisms when agents' valuation functions are drawn from unknown and correlated prior distributions. In particular, we are given a prior distribution $D$, and we are interested in designing a (truthful) mechanism that has good performance for all "true distributions" that are close to $D$ in Total Variation (TV) distance. We show that DSIC and BIC mechanisms in this setting are strongly robust with respect to TV distance, for any bounded objective function $\mathcal{O}$, extending a recent result of Brustle et al. ([BCD20], EC 2020). At the heart of our result is a fundamental duality property of total variation distance. As direct applications of our result, we (i) demonstrate how to find approximately revenue-optimal and approximately BIC mechanisms for weakly dependent prior distributions; (ii) show how to find correlation-robust mechanisms when only ``noisy'' versions of marginals are accessible, extending recent results of Bei et. al. ([BGLT19], SODA 2019); (iii) prove that prophet-inequality type guarantees are preserved for correlated priors, recovering a variant of a result of D{\"u}tting and Kesselheim ([DK19], EC 2019) as a special case; (iv) give a new necessary condition for a correlated distribution to witness an infinite separation in revenue between simple and optimal mechanisms, complementing recent results of Psomas et al. ([PSCW22], NeurIPS 2022); (v) give a new condition for simple mechanisms to approximate revenue-optimal mechanisms for the case of a single agent whose type is drawn from a correlated distribution that can be captured by a Markov Random Field, complementing recent results of Cai and Oikonomou ([CO21], EC 2021).