Given any graph G, the (adjacency) spread of G is the maximum absolute difference between any two eigenvalues of the adjacency matrix of G. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers n, the n-vertex graph G that maximizes spread is the join of a clique and an independent set, with 2n/3 and n/3 vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all n sufficiently large. As an intermediate step, we prove an analogous result for a family of operators in the Hilbert space over L 2 [0, 1]. The second conjecture claims that for any fixed e ≤ n 2 /4, if G maximizes spread over all n-vertex graphs with e edges, then G is bipartite. We prove an asymptotic version of this conjecture. Furthermore, we exhibit an infinite family of counterexamples, which shows that our asymptotic solution is tight up to lower order error terms.