"This talk is devoted to the analysis of the Euler and the Navier--Stokes equations in the context of incompressible fluids. Despite their importance in modelling several natural phenomena, their rigorous mathematical study remains vastly incomplete. Indeed, even though these equations were proposed hundreds of years ago, mayor questions such as existence, uniqueness and smoothness of solutions presently remain extremely challenging open problems. We focus on the uniqueness of solutions to the incompressible Euler equations and on the inviscid limit of solutions to the Navier--Stokes equations. In the class of admissible weak solutions, we can prove a weak-strong uniqueness result for the incompressible Euler equations assuming that the symmetric part of the gradient belongs to $L^1_{\rm loc}([0,+\infty);L^{exp}(R^d;R^{d \times d}))$, where $L^{exp}$ denotes the Orlicz space of exponentially integrable functions. Moreover, under the same assumptions on the limit solution to the Euler system, we can obtain the convergence of vanishing-viscosity Leray--Hopf weak solutions to the Navier--Stokes equations."