After the pioneering work of J. Lurie in [DAG-IX], the possibility of a derived version of analytic geometry drew the attention of several mathematicians. In this talk I will provide several motivations for such a theory, mainly coming from Gromov-Witten theory and shifted symplectic structures. I will then survey the main results of my PhD thesis, including a study of the analytification functor, derived versions of GAGA theorems and the existence of the analytic cotangent complex.