  Menon's identity states that for every positive integer $n$ one has $\sum
(a-1,n) = \varphi(n) \tau(n)$, where $a$ runs through a reduced residue system
(mod $n$), $(a-1,n)$ stands for the greatest common divisor of $a-1$ and $n$,
$\varphi(n)$ is Euler's totient function and $\tau(n)$ is the number of
divisors of $n$. Menon's identity has been the subject of many research papers,
also in the last years. We present detailed, self contained proofs of this
identity by using different methods, and point out those that we could not
identify in the literature. We survey the generalizations and analogs, and
overview the results and proofs given by Menon in his original paper. Some
historical remarks and an updated list of references are included as well.
