In this report, we propose to study some logics and to give some elements of proof theory and models theory. The logics analysed are classic logic, modal and temporal logics, and intuitionist logic. We develop for each of them the formal systems who express them and their associated semantics. Some comparisons will be carry out between those logics in the goal to express the sense of some important notions of logic. We will also see the differences who may appear in mathematics in the context of intuitionist logic. Relations between classic logic and intuitionist logic will be analysed, then some principles and theorems on intuitionist mathematics will be introduced.