Abstract. It is known since Fermat and Euler that prime numbers of the form $4n+1$ are precisely the ones (in addition to 2) which can be written as the sum of two squares. Because of the simplest case of the prime ideal theorem this means that we can count asymptotically the number of primes $p=x^2+y^2$ within a large disc $x^2+y^2 \le X$ in the Euclidean plane. In joint work with Henryk Iwaniec we study some natural generalizations of this question with particular emphasis on analogues concerning points in a large disc in the hyperbplic plane.