In this chapter we describe the concept of minimal living cells, defined as synthetic or semi-synthetic cells having the minimal and sufficient number of components to be endowed with the main biological properties of living cells. The construction of minimal cells starting from isolated compounds is an issue in synthetic biology, origins of life studies, and biotechnology. We start by discussing the different concepts underlining the three above-mentioned fields, by comparing the different viewpoints and highlighting common perspectives. We focus on the first two approaches, firstly describing our recent investigation on the construction of semi-synthetic minimal cells (developed in the Synthcells project), based on the use of liposomes as cell models. A short review of most relevant studies in the field is also given. The emphasis is then shifted to more basic biophysical aspects that emerged from these studies and that can significantly contribute to the understanding of the origins of primitive cells. In particular, we report the unexpected finding of spontaneous self-concentration of proteins and other solutes inside lipid vesicles. This recent discovery gives rise to a several theoretical and experimental implications that are shortly discussed. As a conclusion, we comment on the stateof-the-art in the field, next developments, and future challenges, and highlight how this research may contribute to improve our understanding of life.