The exponential mechanism is a fundamental tool of Differential Privacy (DP) due to its strong privacy guarantees and flexibility.  In this talk I will present a recent extension (joint work with Jordan Awan, Ana Kenney, and Aleksandra Slavkovic from PSU) to settings with summaries based on infinite-dimensional outputs such as with functional data analysis, shape analysis, and nonparametric statistics.  We show that one can design the mechanism with respect to a specific base measure over the output space, such as a Gaussian process.  We provide a positive result that establishes a Central Limit Theorem for the exponential mechanism quite broadly.  We also provide an apparent negative result, showing that the  magnitude of the noise introduced for privacy is asymptotically non-negligible relative to the statistical estimation error. We develop an epsilon-DP mechanism for functional principal component analysis, applicable in separable Hilbert spaces. We demonstrate its performance via simulations and applications to two datasets.