Large sieve inequalities provide powerful notions of "quasi-orthogonality" that are useful in many averaging problems in analytic number theory. The large sieve for Dirichlet characters is essentially optimal in all aspects, but when trying to extend this result to averages over the full universal family of cuspidal automorphic representations of GL(n) (ordered by analytic conductor), the situation is much more difficult for several reasons. From 2000-2005, Brumley, Duke, and Kowalski proved such a large sieve, optimal in the length-aspect, assuming progress towards the generalized Ramanujan conjecture that is only known to hold for the full universal GL(n) family when n is at most 4. I will present joint work with Asif Zaman in which we remove this hypothesis towards the generalized Ramanujan conjecture, proving an unconditional GL(n) large sieve for all n. This leads to the first unconditional zero density estimates for the L-functions in this family. I will also discuss applications to subconvexity bounds.