In this talk we'll consider what can be said about the eigenvalues of A+B, given the eigenvalues of A and the eigenvalues of B. We'll begin by exploring the challenges of this problem and why the answer should look very different depending on whether or not A and B are Hermitian. We will then dive into the world of representation theory and combinatorics for a remarkable complete description of the set of all possible eigenvalues for Hermitian A and B. Finally, now that we know the set of all possible eigenvalues we'll see how free probability can give a sense of what they typically look like even for non-Hermitian A and B.