We discuss recent mathematical constructions of self-similar gravitational collapse for Newtonian stars governed by the Euler-Poisson system, known as Larson-Penston solutions for the isothermal stars and Yahil solutions for polytropic stars, and thereby show the existence of smooth initial data that lead to finite time collapse, characterized by the blow-up of the star density. The talk is based on joint works with Yan Guo, Mahir Hadzic and Matthew Schrecker.