Abstract: Conditions for a local field theory expressed in Hamiltonian form to be Carroll invariant are given.  Electric and magnetic Carrollian limits of Lorentz-invariant theories are then defined and illustrated on various examples.  The electric version of Carroll invariant gravity is also discussed and connected with the Belinski-Lifshitz-Khalatnikov (BKL) behaviour of the gravitational field near a spacelike singularity.  If time permits, asymptotic properties of Carroll gravities will be discussed. Based on M. Henneaux and P. Salgado-Rebolledo, arXiv:2109.06708 [hep-th] and on upcoming work with O. Fuentealba, P. Salgado-Rebolledo and J. Salzer