Lipschitz free p-spaces for p<1 were introduced in [F. Albiac and N. J. Kalton, Lipschitz structure of quasi-Banach spaces, Israel J. Math. 170 (2009), 317–335] to provide the first known example of separable quasi-Banach spaces that are Lipschitz homeomorphic but fail to be linearly isomorphic. The systematic investigation carried out in [F. Albiac, J. L. Ansorena, M. Cúth, and M. Doucha, Lipschitz free p-spaces for 0<p<1, Israel J. Math. 240 (2020), no. 1, 65–98] revealed a series of fundamental obstacles that distinguish the quasi-Banach framework from the classical case p=1. In this talk, we will discuss the recent advances in this area made by the authors.