A liquid droplet placed on a rigid, planar surface has a captivating simplicity: its surface is a spherical cap and remains in equilibrium. In this talk, I’ll show that the addition of different deformable interfaces lead to a range of new phenomena: evaporating droplets with an elastic skin may develop flat tops, while droplets placed in a flexible channel invariably push themselves out. Even without a complex rheology or locomotion, reaching equilibrium can happen slowly, with the droplet’s contact ageing. I will discuss three specific examples, presenting a combination of theory and experiment for each.