In the standard picture of cosmology, the Universe began at the Big Bang; the Big Bang itself is a singularity where the laws of physics break down. A quantum theory of gravity should resolve this singularity and help in understanding the initial state of the Universe needed to account for present observations. I will present some progress towards this goal in the group field theory approach to quantum gravity, using the idea of a universe formed as a "condensate", i.e. a very homogeneous quantum configuration, from a large number of discrete building blocks of geometry. I will show how this setting produces new cosmological models without an initial singularity, and how it can suggest a new mechanism for the generation of cosmological perturbations through quantum gravity vacuum fluctuations.