We give an overview of the theory of framed correspondences in motivic homotopy theory.
Motivic spaces with framed transfers are the analogue in motivic homotopy theory of
-spaces in
classical homotopy theory, and in particular they provide an algebraic description of infinite
-loop
spaces. We will discuss the foundations of the theory (following Voevodsky,
Garkusha, Panin, Ananyevskiy, and Neshitov), some applications such as the
computations of the infinite loop spaces of the motivic sphere and of algebraic
cobordism (following Elmanto, Hoyois, Khan, Sosnilo, and Yakerson), and some open
problems.