Process anonymity has been studied for a long time. Memory anonymity is more recent. In an anonymous memory system, there is no a priori agreement among the processes on the names of the shared registers they access. This article introduces the fully anonymous model, namely a model in which both the processes and the memory are anonymous. It is shown that fundamental problems such as mutual exclusion, consensus, and its weak version called set agreement, can be solved despite full anonymity, the first in a failure-free system, the others in the presence of any number of process crashes.