ABSTRACT: 

We consider a continuous time  Markov chain on a countable space (finite or infinite).

We focus on the  large deviations of the empirical measure, flow and current, as well as their generalization to higher order. In several cases  the associated rate function is explicit. As application  we investigate   the Gallavotti-Cohen functional and its symmetry,  and  discuss   several systems, including birth-and-death Markov chains, random walks on lattices  approximating diffusions under an external potential, particle systems.