The privacy-utility tradeoff problem is formulated as determining the privacy mechanism (random mapping) that minimizes the mutual information (a metric for privacy leakage) between the private features of the original dataset and a released version. The minimization is studied with two types of constraints on the distortion between the public features and the released version of the dataset: (i) subject to a constraint on the expected value of a cost function $f$ applied to the distortion, and (ii) subject to bounding the complementary CDF of the distortion by a non-increasing function $g$. The first scenario captures various practical cost functions for distorted released data, while the second scenario covers large deviation constraints on utility. The asymptotic optimal leakage is derived in both scenarios. For the distortion cost constraint, it is shown that for convex cost functions there is no asymptotic loss in using stationary memoryless mechanisms. For the complementary CDF bound on distortion, the asymptotic leakage is derived for general mechanisms and shown to be the integral of the single letter leakage function with respect to the Lebesgue -- Stieltjes measure defined based on the refined bound on distortion. However, it is shown that memoryless mechanisms are generally suboptimal in both cases.