In contrast to ordinary condensed matter systems, "living systems" are unique. They are based on molecular compartments that reproduce themselves through (i) an uptake of ingredients and energy from the environment, and (ii) spatially and timely coordinated internal chemical transformations. These occur on the basis of instructions encoded in information molecules (DNAs). Life originated on Earth about 4 billion years ago as self-organised systems of inorganic compounds and organic molecules including macromolecules (e.g. nucleic acids and proteins) and low molar mass amphiphiles (lipids). Before the first living systems emerged from non-living forms of matter, functional molecules and dynamic molecular assemblies must have been formed as prebiotic soft matter systems. These hypothetical cell-like compartment systems often are called "protocells". Other systems that are considered as bridging units between non-living and living systems are called "minimal cells". They are synthetic, autonomous and sustainable reproducing compartment systems, but their constituents are not limited to prebiotic substances. In this review, we focus on both membrane-bounded (vesicular) protocells and minimal cells, and provide a membrane physics background which helps to understand how morphological transformations of vesicle systems might have happened and how vesicle reproduction might be coupled with metabolic reactions and information molecules. This research, which bridges matter and life, is a great challenge in which soft matter physics, systems chemistry, and synthetic biology must take joined efforts to better understand how the transformation of protocells into living systems might have occurred at the origin of life.