We study the dynamics of compressible viscous fluids for which capillarity effects cannot be neglected. Their evolution is governed by the Navier-Stokes-Korteweg system, under general power-law assumptions on the capillarity coefficient and the density-dependent viscosity. We provide an overview of some of the main challenges in the analysis of capillary fluids and discuss several questions arising in the one-dimensional setting. In particular, we focus on well-posedness results for strong and weak solutions, as well as relative entropy methods. Should time permit, we will also address questions concerning the long-time behaviour in the stochastic setting. The talk is based on joint work with Stefano Spirito and Donatella Donatelli (University of L’Aquila).