Cellular activity depends to a large extent on membrane bilayer dynamics. Many processes, such as organelle biogenesis and vesicular transport, rely on alterations in membrane structure and shape. It is now widely accepted that intracellular membrane curvature generation and remodelling is mediated and regulated by protein action, and the mechanisms behind the processes are currently being revealed. Here, we will briefly discuss the key principles of membrane deformation and focus on different endocytic events that use various kinds of proteins to shape the plasma membrane into transport carriers. The entry routes are adopted to make sure that a vast variety of molecules on the cell surface can be regulated by endocytosis. The principles for membrane sculpting of endocytic carriers can be viewed either from a perspective of rigid coat budding or of flexible opportunistic budding. We will discuss these principles and their implications, focusing on clathrin-dependent and -independent carrier formation and the proteins involved in the respective pathways.