On the distribution of sums of two squares

The set of sums of two squares is a multiplicative set, studied since Landau and Ramanujan. It has density 0 but is denser than the primes.
In this talk I'll review some classical facts about sums of two squares, such as their asymptotics.
I'll then discuss recent works on the subject (both in the integer setting and the function field setting), and compare to analogous results in primes. I'll focus on various aspects of the sum of a Dirichlet character summed over sums of two squares. In particular, I'll discuss Chebyshev's bias for sums of two squares.