Quantum graphs (Laplacians on metric graphs) play an important role as an intermediate setting between Laplacians on Riemannian manifolds and discrete Laplacians on graphs. Whereas on finite metric graphs the Laplacian is always self-adjoint and has discrete spectrum, the spectral properties of Laplacians on graphs with infinitely many vertices and edges are much less understood. Intuitively, the self-adjointness problem in this case is closely related to finding appropriate boundary notions for infinite graphs. In this talk we study the connection between self-adjointness and the notion of graph ends, a classical graph boundary introduced independently by Freudenthal and Halin. Based on joint work with Aleksey Kostenko (Ljubljana & Vienna) and Delio Mugnolo (Hagen).