Attempts to generalize Helly's theorem to sets of lines intersecting convex sets led to a series of results relating the geometry of a family of sets in R^d to the structure of the space of lines intersecting all of its members. We review recent progress in the special case of disjoint Euclidean balls in R^d, more precisely the inter-related notions of cone of directions, geometric permutations and Helly-type theorems, and discuss some algorithmic applications.