Large-scale, ordered magnetic fields in several astrophysical sources are supposed to be originated, and maintained against dissipation, by the combined amplifying action of rotation and small-scale turbulence. For instance, in the solar interior, the so-called α−Ω mean-field dynamo is known to be responsible for the observed 22-years magnetic cycle. Similar mechanisms could operate in more extreme environments, like proto neutron stars and accretion disks around black holes, for which the physical modelling needs to be translated from the regime of magnetohydrodynamics (MHD) and Newtonian gravity to that of a plasma in a general relativistic curved spacetime (GRMHD). Here we review the theory behind the mean field dynamo in GRMHD, the strategies for the implementation of the relevant equations in numerical conservative schemes, and we show the most important applications to the mentioned astrophysical compact objects obtained by our group in Florence. We also present novel results, such as three-dimensional GRMHD simulations of accretion disks with dynamo and the application of our dynamo model to a super massive neutron star, remnant of a binary neutron star merger as obtained from full numerical relativity simulations.