Rupture and buckling of artificial and biological membranes is an important part of many biological processes. In this review, we present some of the main experimental facts and their analysis. Recent theoretical work, in particular thin film models and nucleation mechanisms of membrane instability, are discussed in detail. Possible applications to membrane adhesion and fusion are pointed out. Attempts are made to explain biological phenomena and experimental results for biological membranes based on a rigorous physicochemical approach developed previously for thin films in colloid systems.