The understanding of the shape-change dynamics leading to the budding and division of artificial cells has gained much attention in the past few decades due to an increased interest in designing stimuli-responsive synthetic systems and minimal models of biological self-reproduction. In this respect, membranes and their composition play a fundamental role in many aspects related to the stability of the vesicles: permeability, elasticity, rigidity, tunability and response to external changes. In this review, we summarise recent experimental and theoretical work dealing with shape deformation and division of (giant) vesicles made of phospholipids and/or fatty acids membranes. Following a classic approach, we divide the strategies used to destabilise the membranes into two different types, physical (osmotic stress, temperature and light) and chemical (addition of amphiphiles, the addition of reactive molecules and pH changes) even though they often act in synergy when leading to a complete division process. Finally, we review the most important theoretical methods employed to describe the equilibrium shapes of giant vesicles and how they provide ways to explain and control the morphological changes leading from one equilibrium structure to another.