$\phi$ and $\sigma$: from Euler to Erd\H os

Euler's $\phi$ function, which counts the number of positive integers relative prime to and smaller than its argument, as well as the sum of divisors function $\sigma$, play an important role in number theory and its applications. In this paper, we survey various old and new results related to the distribution of the values of these two functions, their popular values, their champions, and the distribution of those positive integers satisfying certain equations involving such function, like the perfect numbers and the amicable numbers. In the second part of this paper, we discuss some of the ideas which are used in the proof of a recent result of Ford, Luca, and Pomerance which says that there are infinitely many common values in the ranges of these two functions. This settles a 50 year old question of Erd\H os.