There is now a huge literature on Bayesian methods for variable selection in linear models that use spike-and-slab priors. Such methods, in particular, have been quite successful for applications in a variety of different fields. A parallel methodological development has happened in graphical models, where priors are specified on precision matrices. In this talk I will describe priors for edge selection for the estimation of multiple graphs that may share common features, such as presence/absence of edges or strengths of connections. I will motivate the development of the models using specific applications from neuroimaging and from studies that use large-scale genomic data. If time allows I will also describe extensions of the models to non-Gaussian data and computational challenges.