In this paper we define a concept of equisingularity for families of plane curves in any characteristic. For this, we give a natural generalization of Zariski's equisingularity in the case of plane curves over non algebraically closed fields. This new concept avoids the pathologies in positive characteristic of Zariski's theory. We prove that Zariski's concept is a particular case of ours and we also see that the new one is an open condition.