This talk is divided in two parts. In the first part, we will summarize the
status of the APEmille project that will Le completed by the end of the ye
ar. We will then devote the rest of the talk to the description of a new pr
oject for a multi-TeraFlop machine, apeNEXT. The interested reader will fin
d a much more detailed discussion of all the items touched upon here in the
full proposal of the project that will shortly appear on hep-lat.