Half of the Sun-like stars in our galaxy have a stellar companion. The number of detected planets orbiting binary stars is rapidly increasing thanks to the follow-up of TESS planet candidates by GAIA and direct imaging, which detect stellar companions in systems known to host planets. Important questions regarding the origin and demographics of planets in binaries are starting to be explored, namely, how does the presence of a stellar companion affects the outcome of planet formation? How does the occurrence rate of planets depend on the binary separation? Is there a radius valley for planets orbiting binary stars? In this presentation I will summarise the latest findings about S-type planets (planets orbiting one of the two stars in a binary system), and I will introduce the global planet formation model that we are developing for binary stars. I will show the first results of the population synthesis for S-type planets and discuss the link to the observed population.