Thin polymer sheets floating at fluid interfaces have been a rich setting to study pattern formation and encapsulation. In this talk, I will discuss simpler underlying issues in the attachment of thin elastic sheets to fluid-fluid interfaces.  To better understand the dynamics of attachment, we have carried out experiments in which a polymer sheet is peeled off a liquid surface or draped onto it at a controlled rate.  In the static situation, where the sheet is just held fixed on the liquid, our measurements clear up a confusing issue regarding the stress in a floating sheet and reveal a new class of boundary condition at the edge of the sheet.  When the sheet is being peeled off slowly, we show that the measured forces of peeling provide a new and precise way to obtain contact angles.  With this technique, we are able to access a rich phenomenology of wetting and dewetting hysteresis, including repeatable noise in the motion of a front, and the analogue of pulse duration memory in other depinning systems.