In a paper published posthumously, P.S. Urysohn [3] constructed a complete, separable metric space that contains an isometric copy of every complete separable metric space; it is now referred to as the Urysohn universal space. In this talk, I examine various convexity properties of this special universal space and show that it has a finite ball intersection property even though the Urysohn universal space is not hyperconvex [2]. (This is a joint work with Z. Ibragimov [1]).