  We consider the space $\mathcal{M}$ of Euclidean similarity classes of framed
loops in $\mathbb{R}^3$. Framed loop space is shown to be an
infinite-dimensional K\"{a}hler manifold by identifying it with a complex
Grassmannian. We show that the space of isometrically immersed loops studied by
Millson and Zombro is realized as the symplectic reduction of $\mathcal{M}$ by
the action of the based loop group of the circle, giving a smooth version of a
result of Hausmann and Knutson on polygon space. The identification with a
Grassmannian allows us to describe the geodesics of $\mathcal{M}$ explicitly.
Using this description, we show that $\mathcal{M}$ and its quotient by the
reparameterization group are nonnegatively curved. We also show that the planar
loop space studied by Younes, Michor, Shah and Mumford in the context of
computer vision embeds in $\mathcal{M}$ as a totally geodesic, Lagrangian
submanifold. The action of the reparameterization group on $\mathcal{M}$ is
shown to be Hamiltonian and this is used to characterize the critical points of
the weighted total twist functional.
