I will discuss basic features of a formulation of higher-spin field theory in which conventional space-time gets extended to a hyperspace with a number of extra dimensions that effectively describe the spin degrees of freedom of the fields in the ordinary space-time. In this formulation an infinite number of higher spin fields are packed into a single scalar and spinor field propagating in the hyperspace. The dynamics of higher spin fields is encoded in equations of motion of the scalar and spinor hyperfields. The hyperfield equatons on flat and AdS-like hyperspaces are related to each other by a generalized conformal transformation, which also relates two-, three- and four-point functions in the AdS-like hyperspace to the corresponding correlators in the flat hyperspace.