A pseudocircle is a simple closed curve in the plane. An intersecting arrangement of pseudocircles is a finite collection of pseudocircles so that any two intersect in exactly two points where they cross. Grünbaum conjectured in the 1970's that in the case of simple arrangements there are at most 2n - 2 digon cells, i.e. cells which have exactly two crossings on its boundary. I will present a result by Agarwal et al. (2004) which proves this conjecture for the special case of cylindrical arrangements. Based on that we show that the conjecture also holds whenever the arrangement contains three pseudocircles which pairwise form a digon cell. Moreover, I will present a result concerning the number of triangles in digon free arrangements, which disproves another conjecture by Grünbaum. (joint with S.Felsner and M.Scheucher)