Helly's theorem of 1923 opened a large field of inquiry now designated as geometric transversal theory. A typical concern is the study of all k-planes (also called k-flats) that intersect all elements of a given family of subsets (or objects) in Rd. These are the k-transversals of the given family and they define a certain subspace of the corresponding Grassmannian. I will present recent results on Helly-type theorems for the existence of line transversals to disjoint balls obtained in collaboration with Ciprian Borcea, Otfried Cheong, Andreas Holmsen and Sylvain Petitjean.