Summary: The main goal of this paper is to study analytically the families of symmetric periodic orbits of the elliptic Sitnikov problem for all values of the eccentricity in the interval \([0,1)\), providing qualitative and quantitative information on the bifurcation diagram of such families of periodic orbits. The basic tool for proving our results is the global continuation method of the zeros of a function depending on one parameter provided by Leray and Schauder and based in the Brouwer degree.