In this document, we present the CATS system. The main ideas behind the design of CATS were to have a general purpose simulator able to be a stable basis for the development of other modules, representing different levels of Air Traffic Control and Air Traffic Management. In the first part of this paper, the basic framework and hypothesis of the CATS simulator are introduced. The second part deals with the general mathematical concepts regarding conflict resolution and airborne collision avoidance systems. Different results obtained so far with modules regarding conflict resolution, avoidance, or workload, built on top of the CATS simulator are presented as examples in the last part.