This article addresses election in fully anonymous systems made up of $n$ asynchronous processes that communicate through atomic read-write registers or atomic read-modify-write registers. Given an integer $d\in\{1,\dots, n-1\}$, two elections problems are considered: $d$-election (at least one and at most $d$ processes are elected) and exact $d$-election (exactly $d$ processes are elected). Full anonymity means that both the processes and the shared registers are anonymous. Memory anonymity means that the processes may disagree on the names of the shared registers. That is, the same register name $A$ can denote different registers for different processes, and the register name $A$ used by a process and the register name $B$ used by another process can address the same shared register.