  This paper has two parts. The first part surveys Euler's work on the constant
gamma=0.57721... bearing his name, together with some of his related work on
the gamma function, values of the zeta function and divergent series. The
second part describes various mathematical developments involving Euler's
constant, as well as another constant, the Euler-Gompertz constant. These
developments include connections with arithmetic functions and the Riemann
hypothesis, and with sieve methods, random permutations and random matrix
products. It includes recent results on Diophantine approximation and
transcendence related to Euler's constant.
