Lipid bilayer membranes play a fundamental role in cell biology serving diverse roles ranging from compartmentalising cells into distinct regions across which chemical gradients are maintained to serving as scaffolds on which many proteins anchor and aggregate. Understanding basic features of these complex soft materials presents many interesting problems and challenges. Lipids are amphilic molecules having hydrophobic–hydrophilic domains and exhibit intriguing phases and collective behaviours such as self-assembly into fluidic structures such as micelles, vesicles, and membrane sheets. In the dynamic responses of these materials, hydrodynamics of the surrounding solvent often plays an important role in coupling both motions of the individual lipids of the bilayer and the lateral motions of protein inclusions. In this talk we shall address primarily this hydrodynamic aspect and related phenomena by introducing new mathematical and computational approaches for the study of lipid bilayer membrane sheets and vesicles. In particular, we shall present results for a continuum mechanics description of lipid bilayer membranes that accounts for how protein inclusions alter local material properties taking into account the bi-directional membrane–protein coupling, solvent hydrodynamics, and membrane thermal undulations. At the molecular level, we shall discuss a new dynamic coarse-grained modelling approach for lipids that treats the solvent implicitly but incorporates the important roles played by hydrodynamics and thermal fluctuations through a numerical fluctuating hydrodynamics method. We will then present specific results of this approach for lipid bilayer membrane sheets and vesicles.