Given some dimension d, what is the maximum number of lines in Rd such that the angle between any pair of lines is constant? (Such a system of lines is called "equiangular".) This classical problem was initiated by Haantjes in 1948 in the context of elliptic geometry. In 1966, Van Lint and Seidel showed that graphs could be used to study equiangular line systems.
Recently this area has enjoyed a renewed interest due to the current attention the quantum information community is giving to its complex analogue. I will give an introduction to the area and report on some new developments in the theory of equiangular lines in Euclidean space. Among other things, I will present a new construction using real mutually unbiased bases (orthonormal bases such that the angle between two elements of different bases is arccos(1/√d)), as well as improvements to two long-standing upper bounds for equiangular lines in dimensions 14 and 16.