Two Correlator sub-systems have been constructed for the ALMA project, one for the Main Array of 12-m antennas and one for the ALMA Compact Array (ACA). Both sub-systems combine the astronomical signals captured by the ALMA antennas to form the images which will be interpreted and modeled by the astronomers. These correlators also have the ability to analyze the spectral content of the incoming radiation ; in particular, they allow us to identify or discover the molecular and atomic species present in the nearby or distant cold Universe where new generations of stars are being formed. The ALMA Correlators can be seen as highly specialized 'super-computing' machines operated with no hard disks at the highest site ever used for astronomical programs. The 17 peta-operations per second performed by the Main Array Correlator may be compared with the fastest and latest generation of super computers operating in the petaflop domain. In this article we briefly introduce the basic principles of correlation and outline some of the architectural differences between the Main Array and ACA Correlators. We present the main technical characteristics of the Main Array Correlator and give some details on its observing modes and performance. Finally, we summarize the present status of these two powerful Correlator sub-systems and recall that several groups across the world were involved in their construction.Read less <