Abstract: Elliptic cohomology is a type of generalised cohomology theory related to elliptic curves which was introduced in the late 1980s. An important motivation for its introduction, which came from physics, was to help understand index theory for families of differential operators over free loop spaces. Yet for a long time, the only known constructions of elliptic cohomology were purely algebraic, and the precise connection to free loop spaces remained obscure. In this talk, I will summarise two constructions of complex analytic, equivariant elliptic cohomology: one from the K-theory of free loop spaces, and one from the ordinary cohomology of double free loop spaces. If time permits, I will also describe the construction of a Chern character-type map from the former to the latter.