The theory of Mirkov\'c-Vilonen (MV) cycles associated to a complex reductive group $G$ has proven to be a rich source of structures related to representation theory. I investigate double MV cycles, which are analogues of MV cycles in the case of an affine Kac-Moody group.
I will shortly review some aspects of the theory of MV cycles for finite-dimensional groups. The story gives rise to MV polytopes and a surprising connection with Lusztig's canonical basis. Then I will discuss double MV cycles. Here the finite-dimensional story does not naively generalize. Nonetheless, in type A, I will present a method to parameterize double MV cycles. This method gives rise to exactly the combinatorics of the Naito-Sagaki-Saito crystal. If I have time, I will discuss some related work and some open problems.