The principles of supersymmetry, originally describing the symmetry between bosons and fermions, provide us with a mathematical tool to reshape arbitrary potentials without losing their symmetry, a concept known as isospectrality. Here, we demonstrate a new transformation paradigm in optics and acoustics based on the invariance of the eigenspectra of the Hamiltonian of a physical system, enabled by supersymmetry. First, by creating an optical gradient-index metamaterial to control local index variations in a family of isospectral optical potentials, we demonstrate broadband continuous supersymmetric transformations in optics on a silicon chip. This allows us to simultaneously transform the transverse spatial characteristics of multiple optical states, enabling arbitrary steering and switching of light flows. Second, by constructing a rigid acoustic material capable of realizing spatially varying indices of airborne sound, we show that the scattering of sound waves through the metamaterial can be effectively controlled using supersymmetry. Through a novel synergy of symmetry physics and metamaterials, our work provides an adaptable strategy to conveniently manipulate the flow of both light and sound while fully exploiting their spatial degrees of freedom.