Fluctuating hydrodynamic descriptions provide a promising approach for modelling and simulating elastic structures that interact with a fluid when subject to thermal fluctuations.  This allows for capturing simultaneously such effects as the Brownian motion of spatially extended mechanical structures as well as their hydrodynamic  coupling and responses to external flows.  A significant advantage of this approach over alternative methods is the ability to handle the hydrodynamic equations directly using spatially adaptive discretizations or using domains having complex geometries.  However, this presents the challenge of numerically approximating a set of stochastic partial differential equations whose solutions are non-classical and only defined in the generalised sense of distributions.  We introduce stochastic discretization procedures based on ideas from statistical mechanics and we show how efficient stochastic computational methods can be developed.  We demonstrate our methods in the context of applications including the simulation of particles within microfluidic devices and the rheological responses of soft materials.  We also survey the current challenges in this field and opportunities for developing new more scalable algorithms.