The ferromagnetic Ising spin model is often used to model second-order phase transitions and the continuous emergence of order. We consider this model on a random graph, where the additional randomness provided by the graph gives a rich picture with a host of surprises. We identify similarities and differences between the quenched and annealed Ising model. We find that the annealed critical temperature is highly model-dependent, even in the case of graphs that are asymptotically equivalent (such as different versions of the simple Erdös-Rényi random graph). The quenched critical temperature is instead the same for all locally tree-like graphs. Moreover, in the presence of inhomogeneities that produce a fat-tail degree distribution, the difference between quenched and annealed becomes even more substantial, leading in some cases to different universality classes and different critical exponents. The annealed properties depend sensitively on whether the total number of edges of the underlying random graph is fixed, or is allowed to fluctuate. If time allows preliminary results on the annealed Potts model, displaying a first-order phase transition, will also be discussed. [This talk is based on several joint works with Hao Can, Sander Dommers, Claudio Giberti, Remco van der Hofstad and Maria Luisa Prioriello. The preliminary work on Potts models also involves Neeladri Maitra and benefited from discussions with Guido Janssen.]