The past decade has seen a pronounced growth of interest in differential operators defined on metric graphs, commonly known as quantum graphs. They offer the technical simplicity of onedimensional objects, while often displaying complex behaviour akin to higherdimensional ones, and are thus particularly useful as toy models. Starting with a brief introduction to the variational analysis and spectral theory of differential operators on graphs, we will present some recent results on eigenvalue estimates for operators of Laplacian and pLaplacian type obtainable by variational means. This is partly based on joint work with Gregory Berkolaiko (Texas A&M), Pavel Kurasov (Stockholm) and Delio Mugnolo (Hagen).
