Title: On the Universality of Langevin Diffusion for Private Euclidean (Convex) Optimization. Abstract: In this paper, we revisit the problem of differentially private empirical risk minimization (DP-ERM) and differentially private stochastic convex optimization (DP-SCO). We show that a well-studied continuous time algorithm from statistical physics, called Langevin diffusion (LD), simultaneously provides optimal privacy/utility trade-offs for both DP-ERM and DP-SCO, under $\epsilon$-DP, and $(\epsilon,\delta)$-DP both for convex and strongly convex loss functions. We provide new time and dimension independent uniform stability properties of LD, with which we provide the corresponding optimal excess population risk guarantees for $\epsilon$-DP.  An important attribute of our DP-SCO guarantees for $\epsilon$-DP is that they match the non-private optimal bounds as $\epsilon\to\infty$. 