It is a remarkable and nontrivial fact that there is a duality on the set of compact connected Lie groups; this interchanges, for example, the group of 2x2 unitary matrices of determinant 1, and the group of rotations in three real dimensions. This duality emerged in mathematics in the 1960s ("Langlands duality"), and independently in physics in the 1970s ("electric-magnetic duality").    I will give a gentle introduction to this duality, without assuming background in Lie theory, and say a few sketchy words about the role it plays in various settings.  I will then say a little about  my recent work with Ben-Zvi and Sakellaridis where we seek  to incorporate into the duality spaces upon which the groups act.