We consider the global rigidity problem for bar-joint frameworks where each vertex is constrained to lie on a particular line in R^d. In our setting we allow multiple vertices to be constrained to the same line. Under a mild assumption on the given set of lines we give a complete combinatorial characterisation of graphs that are generically globally rigid in this setting. This gives a d-dimensional extension of the well-known combinatorial characterisation of 1-dimensional global rigidity. This is joint work with Fatemeh Mohammadi, Harshit J Motwani, Anthony Nixon and Shin-ichi Tanigawa