We discuss in this paper ground-state and nonequilibrium properties of ultracold atoms in optical lattices in the strongly correlated limit. We review fermionic Mott-insulators studied on the basis of quantum Monte Carlo simulations, where local quantum criticality is displayed in one dimension. We continue with exact results for hard-core bosons in one dimension, showing their universal properties in equilibrium, and their nonequilibrium dynamics. Here we show that starting from a Fock state, a quasi-condensate emerges at finite momentum during free expansion. On the other hand, the free evolution of an initially confined quasi-condensate of hard-core bosons leads to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium.